On Preserving: Essays on Preservationism and Paraconsistent Logic (Toronto Studies in Philosophy)

ACKNOWLEDGEMENTS

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Acknowledgements This book owes a great deal to a great many people. So many indeed that we have probably forgotten some of them and to those we apologize in advance. First, since the work reported in this volume sprang from some initial work of Ray Jennings and Peter Schotch, the Social Science and Humanities Research Council of Canada deserves a particularly hearty vote of thanks. The Council made that collaboration possible under a number of operating grants over those first crucial years. Jennings and Schotch had a chance not only to meet face to face, but also to make their work known to a larger audience by giving talks and attending conferences over much of the learned world. Without that support, it is hard to see any of this having happened. Second, and nearly tied for first, we must thank our students over the years. They have been many but we owe them a great debt not only for their enthusiasm and hard work, but also because many of them were responsible for proving things that needed proving. At that head of that list is Bryson Brown, a one-time research assistant who rose to become a famous philosopher. Along the way, he with Peter Apostoli proved the first representation theorem and many graph theoretic results which do not appear in this volume.1 Apart from Bryson the names Blaine D’Entremont, [fill in] were among the early adopters of our ideas and each has made a signal contribution. The latest generation includes Dorian Nicholson, Darko Sarenac and Gillman Payette. Two of those have contributed to this volume and all have contributed to the project in ways too valuable to describe in a paragraph. It also goes without saying that we owe an enormous debt of gratitude to our wives and families who have had to put up with us while we searched for a proof or even just a way of explaining something that seemed clear to us alone.

1 But which will appear in a projected companion volume focused on applications of the preservationist approach.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction to the Essays P ETER S CHOTCH 1.1 The Origins of Preservationism . . . . 1.2 Paraconsistency and Modal Semantics 1.3 The Concept of Level . . . . . . . . . 1.4 Preservation . . . . . . . . . . . . . . 1.5 The Essays . . . . . . . . . . . . . . .

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3 3 7 10 11 12

2 Paraconsistency: Who Needs It? R AY J ENNINGS AND P ETER S CHOTCH 2.1 Introduction . . . . . . . . . . . . . 2.2 The Strange Case of C. I. Lewis . . . 2.3 The Inconsistency of Belief . . . . . 2.4 Inconsistency and Ethics . . . . . . . 2.5 Summing Up . . . . . . . . . . . . .

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15 15 18 21 26 28

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31 31 35 38 39

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3 Weakly Additive Algebras and a Completeness Problem A LASDAIR U RQUHART 3.1 Introduction and Brief History . . . . . . . . . . . . . 3.2 Weakly additive operators . . . . . . . . . . . . . . . 3.3 A Further Generalization . . . . . . . . . . . . . . . 3.4 Duality theory for weakly additive operators . . . . . vi

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4 A Dualization of Neighborhood Structures D ORIAN N ICHOLSON 4.1 Introduction . . . . . . . . . . . . . . . 4.2 Hyperframes . . . . . . . . . . . . . . . 4.3 Normal Hyperframes . . . . . . . . . . . 4.4 Weakly Aggregative Modal Logic . . . .

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47 47 48 52 55

5 Polyadic Modal Logics and Their Monadic Fragments K AM S ING L EUNG AND R.E. J ENNINGS 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 5.2 Polyadic modal languages and multi-ary relational frames 5.3 Normal polyadic modal logics . . . . . . . . . . . . . . 5.4 Frame definability . . . . . . . . . . . . . . . . . . . . 5.5 Soundness and completeness . . . . . . . . . . . . . . 5.6 Diagonal fragments of normal polyadic modal logics . .

59 59 62 65 71 73 78

6 Preserving What? G ILLMAN PAYETTE AND P ETER S CHOTCH 6.1 Introduction . . . . . . . . . . . . . . . . 6.2 Making A Few Things Precise . . . . . . . 6.3 What’s Wrong With This Picture? . . . . . 6.4 Speak of the Level . . . . . . . . . . . . . 6.5 Level Preservation . . . . . . . . . . . . . 6.6 Yes, But Is It Inference? . . . . . . . . . . 6.7 Other Level-preserving Relations . . . . .

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81 81 84 88 89 92 97 99

7 Preserving Logical Structure G ILLMAN PAYETTE 7.1 Introduction . . . . . . . . . . . . 7.2 Definitions . . . . . . . . . . . . . 7.3 Substitutions . . . . . . . . . . . . 7.4 Algebraic Concerns . . . . . . . . 7.5 Denials . . . . . . . . . . . . . . . 7.6 Forcing and Preservationism . . . . 7.7 Structural Rules and Compactness 7.8 First-Order Logic and Forcing . . .

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101 101 104 113 117 121 123 124 128

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CONTENTS

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7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 137 8 Representation of Forcing D ORIAN N ICHOLSON AND B RYSON B ROWN 8.1 Introduction . . . . . . . . . . . . . . . . 8.2 Background . . . . . . . . . . . . . . . . 8.3 Representation . . . . . . . . . . . . . . . 8.4 More  Definitions and motivation . . . . . 8.5 €  `X  . . . . . . . . . . . . . . . . . . 8.6 € X  . . . . . . . . . . . . . . . . . . 8.7 The Representation of Forcing . . . . . . .

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139 139 140 142 146 151 152 152

9 Forcing and Practical Inference P ETER S CHOTCH 9.1 Introduction . . . . . . . . 9.2 Dirty Hands . . . . . . . . 9.3 †-Forcing . . . . . . . . . 9.4 A-Forcing . . . . . . . . . 9.5 Wrap-up . . . . . . . . . .

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155 155 157 158 164 166

10 Ambiguity Games and Preserving Ambiguity Measures B RYSON B ROWN 10.1 Ambiguity . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Logic of Paradox . . . . . . . . . . . . . . . . . 10.3 Restoring Symmetry . . . . . . . . . . . . . . . . . . 10.4 Another Approach to FDE . . . . . . . . . . . . . . . 10.5 Ambiguity and quantification . . . . . . . . . . . . . 10.6 Echoes of supervaluation . . . . . . . . . . . . . . . 10.7 Final Remarks on Preservation . . . . . . . . . . . .

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167 168 169 172 174 179 179 180

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Nomenclature

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References

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Index 193 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . 197

ON PRESERVING

One

Introduction to the Essays P ETER S CHOTCH

The essays in this volume are intended to serve as an introduction to a research project that started a long time ago, back in the 20th Century. But it didn’t start as a project about paraconsistent logic or about what we have come to call preservationism.

1.1

The Origins of Preservationism

Instead it started in the summer of 1975, at Dalhousie University, when Ray Jennings1 and Peter Schotch decided to write a ‘primer’ of modal logic—a project that never quite made it into the dark of print. So it was modal and decidedly not paraconsistent logic which was uppermost in the minds of Jennings and Schotch. In this connection one should keep in mind that neither had so much as heard the word ‘paraconsistent’ at the time we are considering.2 1 Jennings was then (and now) a member of the philosophy department at Simon Fraser University, but he had accepted a summer-school job at Dalhousie in order to work with Schotch. 2 In fact the first time they heard that word was at the 1978 meeting of the Society for Exact Philosophy, held in Pittsburgh. And in all truth, they paid little enough attention to the word the first time they did hear it.

3

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CHAPTER 1. INTRODUCTION TO THE ESSAYS

What got the ball rolling, was the discovery by Jennings, which he communicated to Schotch, that so-called normal modal logic was not suitable for certain philosophical applications of a deontic cast. In particular, there seemed to be distinctions which one might draw intuitively that evaporate in normal modal logic.3 Now the only restriction on the kind of modal logic involved was that it be normal, so enabling distinctions which cannot be made in that logic was not going to be a simple matter of dropping some modal axioms or, on the semantic side, relaxing a frame condition. If we are talking about normal modal logic in general, then there are no frame conditions to relax. The fundamental normal modal logic, often called K, is determined by the class of all frames. Period. But later that afternoon, it started to seem that perhaps the semantics of normal modal logic did impose some restrictions after all. To begin with, there was the restriction that the universe of the frame, its set of ‘possible worlds,’ must be non-empty. That didn’t seem a very promising condition to relax, though there certainly have been those who were prepared to give it up, if only to see what happens to modal logic.4 No, it would have to be a condition on the frame relation which did the trick. Which condition was it? There can be only one, and that is the condition that the frame relation be binary. And there it was. Jennings and Schotch then began trying to work out what the truthcondition for necessity would have to be when the frame relation was not binary but say, ternary. Later that evening they had narrowed it down to only one possibility: ˛ is true at a point x in a model M if and only if for every ordered pair hy; zi such that Rxyz, either ˛ is true at y or ˛ is true at z. The generalization to n-ary frame relations is trivial. 3 The particular example mooted by Jennings had to do with two modal formulas, one of which had a necessity operator outside of a conditional and the other having the necessity distributed over the conditional. Perhaps the easiest version of this kind of distinction, and the one which became absolutely central to the program, is the one between ˛  ::˛ and :?. The latter claims that no contradiction can be necessary while the former asserts that if any formula is necessary, its negation is not. 4 Charles Morgan has constructed this kind of modal logic.

1.1. THE ORIGINS OF PRESERVATIONISM

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In the hot flush of discovery, Jennings and Schotch imagined that they were treading on terra incognita, but as Goethe reminds us, All the great thoughts have already been thought. Our job is merely to think them again. n-ary frames had already been introduced by Bjarni J´onsson and Alfred Tarski (Tarski and J´onsson, 1951). In that work n-ary frames are a way of representing Boolean Algebras with n-1-ary operators. Modal logic following Kripke had rediscovered (a decade after the original discovery) the special case of this idea, for binary frames. The problem for Jennings and Schotch was how to axiomatize a fragment of the Tarski and J´onsson n-ary frames, the diagonal fragment. In other words, in the original work the extra operator is (n-1)-ary, let’s call it
. So for every n-1-tuple of formulas ˛1 ; : : : ; ˛n 1 the ‘modal’ formula is
.˛1 ; : : : ; ˛n 1 /. This notion, as interesting and useful as it no doubt is, isn’t the modal notion selected by Jennings and Schotch. Theirs, the unary ˛, is represented in Tarski and J´onsson’s terms as
.˛; : : : ; ˛/ where the . . . represents n-3 iterations of the formula ˛. This was not a problem which had ever been considered before so Jennings and Schotch were happily refuting Goethe, or at least attempting to do that. The thing is: Using the resources of normal modal logic, the problem is extremely difficult. If we stick to the ternary case for ease of exposition, one quickly finds that the principle of complete modal aggregation [K] ` ˛ ^ ˇ  .˛ ^ ˇ/ doesn’t hold, although the rule of monotonicity (sometimes called regularity) and the rule of (unrestricted) neccessitation (also called the rule of normality): [RM] ` ˛  ˇ H) ` ˛  ˇ [RN] ` ˛ H) ` ˛ both hold in ternary frames (and indeed in n-ary frames).

CHAPTER 1. INTRODUCTION TO THE ESSAYS

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So now comes the problem: How do you axiomatize this semantics? It turns out to be quite a trick without [K]. In fact, it turns out to be impossible without some kind of aggregation principle, as Jennings and Schotch realized early on in the project. The correct version of aggregation seemed to be the principle: [K3] ` .˛1 ^ ˛2 ^ ˛3 /  ..˛1 ^ ˛2 / _ .˛2 ^ ˛3 / _ .˛1 ^ ˛3 // But how to prove this conjecture? The situation is entirely different in K, where all the instances of the principle of complete modal aggregation that you might need follow transparently from [K]. In the general case, one needs to show that every aggregation principle which holds in virtue of ‘pigeonhole reasoning’5 , (the way [K3] does) follows from [K3]. That is a non-trivial piece of combinatorial work. While pondering this question and writing a few papers on modal aggregation (none of which solved the problem), Jennings and Schotch chanced to attend the meeting of the Society for Exact Philosophy in Pittsburgh which was held in June of 1978. It was there that they first heard of paraconsistent logic via a paper on the subject presented by Robert Wolf. It was also there that they first saw normal modal logic axiomatized by the rule: [N] € ` ˛ H) Œ€ ` ˛ (where Œ€ stands for f j 2 €g) when it was used in a paper presented by Brian Chellas.6 In the question period following the presentation by Jennings and Schotch, Barbara Partee happened to remark that it all seemed to her like an attempt to represent a kind of non-trivial reasoning from inconsistent data, since, without complete modal aggregation one cannot construct ? from ˛ and :˛. So in particular, if one thought of the  as some sort of belief operator then . . . . Jennings and Schotch took note of 5 By

this is meant merely the principle that if there are k objects to distribute over k containers, at least one container must contain at least two objects 6 Chellas later attributed the rule to Dana Scott.

1

1.2. PARACONSISTENCY AND MODAL SEMANTICS

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how the audience perked up at this, and kindly thanked Professor Partee for the observation. It was on the flight back to Halifax7 that Jennings urged Schotch to consider the Chellas axiomatization in connection with their n-ary modal logic. What Jennings had seen is that the rule [N] characterizes necessity in terms of some notion of inference.8 For the necessity of normal modal logic in the usual sense, the ‘corresponding’9 inference relation is simply the classical one. But Jennings presumed, since the nary semantics lacked complete modal aggregation, the inference relation that characterized the new notion of necessity cannot be classical. This turned out to be an absolutely crucial insight, and one which led ultimately to the program now called preservationism. Schotch began trying to discover an inference relation which bears the same relation to n-ary modal logic as the classical relation bears to normal modal logic. There will of course be infinitely many such relations corresponding to the different orders of the frame relations. The project didn’t take long since it was possible to, in effect, read off the relation from the modal semantics.

1.2

How Parconsistent Inference Fell Out of Modal Semantics

The trick is to begin as though you already have the relation in question and then start the ‘hard’ direction of the fundamental theorem for all, let’s say, ternary modal logics. These will be all the extensions of the base logic K3 which logic is axiomatized by the single rule (on top of classical sentence logic): [N3] € B3 ˛ H) Œ€ ` ˛ In order to discover just what properties B3 has to have, we see what will be needed to prove: 7 Where

Jennings was teaching summer school again. algebraist would say that the two concepts inference and necessity, are Galois connected. 9 Depending upon how precise one wishes to make the notion of correspondence used here, one might be restricted to the necessity operator of the logic K, rather than the necessity operator of any normal modal logic. 8 An

CHAPTER 1. INTRODUCTION TO THE ESSAYS

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˛ 62 † H) hMK3 ; †i ² ˛ Where MK3 is the K3 canonical model and † is an element of the K3-canonical domain, which is to say a maximal K3-consistent set of formulas. The converse of this is trivial so long as we define the K3-canonical relation, RK3 over the elements of K3-canonical domain, in the obvious way as: RK3 €1 2 ” .8˛/.˛ 2 € H) ˛ 2 1 or ˛ 2 2 / So assume ˛ 62 †. We need to find a pair of maximally K3consistent sets to which † is related, neither of which contains ˛. The existence of such a pair amounts to the failure of ˛ at the element † of the K3-canonical model, which is what we are trying to show. To show that a pair of maximally K3-consistent sets exists to which † stands in the canonical relation, it suffices to show that there is some partition of the set .†/ D fˇjˇ 2 †g both cells of which are K3-consistent. In such a case each cell can be extended to maximal K3-consistency in the usual way. We also require that ˛ not belong to either of the maximally K3-consistent sets thus constructed. The only way make sure of this is to add :˛ to each cell of the partition. From this point we argue by reductio: Suppose that no such partition can be found, which is to say that for every partition of .†/ into two K3-consistent cells, at least one of the cells is not K3-consistent with :˛. By classical reasoning, this amounts to saying that at least one cell of every partition of .†/ into two K3-consistent sets (classically) proves ˛. If our axiomatization of K3 is going to work, what we just wrote must amount to a statement that .†/ B3 ˛. This is because if and only if the inference relation is defined in that particular way then, by appeal to the rule [N3], we must have: Œ.†/ ` ˛. But the set on the left of ` is a subset of † since it collects those formulas in † which are ‘guarded’ by . More precisely, the first (inner) operation forms the set of those formulas in † which begin with  and strips off the leading  while the second (outer) operation prefixes

1.2. PARACONSISTENCY AND MODAL SEMANTICS

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each of those formulas with . This operation leaves us with exactly those formulas in † which have an initial . This subset will usually be proper, since an arbitrary set like † will normally contain lots of formulas ˛ which do not begin with  such that ˛ does not belong to †. Now if a subset of † proves ˛ then † itself must likewise prove that formula by monotonicity of inference.10 But, by classical reasoning, every maximally K3-consistent set is a theory, which is to say that it contains every formula that it proves. So ˛ 2 †, which contradicts the hypothesis. Thus K3 is determined by the class of all ternary frames. In order for B3 to be a paraconsistent inference relation, it must be an inference relation. There is cheerful news on that matter: The relation satisfies all the classical structural rules. So yes, it is inference of some kind, but is it really paraconsistent? Well, yes and no. It is at least a partially paraconsistent relation, since, if the premises set contains only two contradictory premises without either being self-contradictory, then the closure of the set under the relation B3 will not contain every formula. On the other hand the premise set might contain a subset like: f˛1 ; :˛1 ^ ˛2 ; :˛1 ^ :˛2 g for which any individual member is consistent by itself but each is inconsistent with the other two. Any set containing a subset like this cannot be partitioned into only two consistent cells. So, contrary to the spirit of paraconsistency perhaps, all such sets prove (in the sense of B3 every formula. So B3 allows sets to explode inferentially even though they may be representable as the union of sets all of which are consistent. Elsewhere this condition is vilified by branding it as a kind of inconsistency.11 Now, of course we can move to a relation defined in terms of partitions containing three cells. In that situation there will be an inference relation, call it B4 which does what we want but it will, in turn, fall 10 This

is, in effect, an appeal to the classical structural rule of dilution.

11 In (Schotch and Jennings, 1989) inference relations which allow sets to explode without their

containing an absurd formulas are called inconsistent*.

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prey to sets of 4 formulas each consistent in isolation but explosive in combination with any of the others.12 Are we to be driven to infinity here? No, but before we see why not, let us come clean. There are premise sets such that no partition into consistent cells (even allowing infinite partitions) is possible. These are the sets that contain formulas which are, by themselves, inconsistent. Such formulas are often called absurd, with ˛ ^ :˛ serving as the most popular classical example. Those inconsistent sets, the ones containing absurd formulas, cannot be fixed. No matter which version of our new relation we take, the consequences of such a a set under that relation will be the set of all formulas. If the goal of paraconsistent logic is prevent even absurd formulas from exploding, then our relations are, none of them, paraconsistent. Fortunately, neither Jennings nor Schotch (nor the angels, we are tempted to say) believe that paraconsistent logic has that goal. Thus was a great gulf fixed between the preservationist approach, and the ‘dialethic’ approach championed by several people in the Southern Hemisphere.13 It is characteristic of the latter view that both a sentence and its negation might be true, and that there could be a way of making ˛ ^:˛ true while at the same time making ˇ false. Hence, even absurd formulas might be redeemed. It is tempting to refer to this program as ‘non-negative’ on the ground that whatever else may be going on, if you succeed in making both ˛ and :˛ true, then you must be using the symbol : to represent something other than negation. This temptation is reinforced by the fact that the dialethists insist on referring to the preservationist program as ‘nonadjunctive.’

1.3

The Concept of Level

In the previous section it seemed as though there was no limit upon how large we might have to make a partition in order to ensure that every premise set (not containing an absurdity) could be partitioned into that 12 B

4 and the earlier B3 are examples of what 13 The name Graham Priest springs to mind.

we call fixed-level forcing relations.

1.4. PRESERVATION

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many consistent cells. That is no doubt true but not entirely to the point. What we clearly need to do is to treat every premise set on its own merits, rather than assuming that there is some fixed number n such that the premises can be partitioned into no more than n consistent cells. Rather, let us say that the level of a set, say € of formulas, modulo some technicalities which will be explained in detail below, is the least number of cells (if there is such a thing) such that € can be partitioned into that many consistent cells. If there is no such number, which is to say if € contains an absurd formula, then assign the symbolic value 1 to be the level. And now we have what we might regard as a whole-hearted paraconsistent inference  relation. We refer to this relation as the forcing relation indicated by and it is defined not in reference to some fixed number of cells, but to partitions into the number of consistent cells equal to the level of the premise set.  If, on every such partition, at least one of the cells proves ˛, then € ˛.

1.4 Preservation At this point the modal logical motivation has fallen away. In the first place, the version of the generalized form of the rule [N], which has forcing on the left of H) , doesn’t correspond to any particular n-ary modal semantics, but rather to a semantics in which the frame relation is allowed to vary. This is a bit of an odd duck, modally speaking. In the second place, the problem which started the whole thing off, that of showing that the n-ary semantics can be axiomatized using [RN], [RM], and the appropriate n-ary form of the aggregation principle [KN] is still unsolved!14 So where does preserving come in? During several presentations that Jennings and Schotch made in the early 1980’s more than once the complaint was heard that the notion of inference based on forcing and its near relations seemed at odds with the ‘truth-preservation’ account.15 14 As a matter of historical fact, several years had to pass before that problem was finally solved. See chapter 4 in this Volume. 15 This paradigm is considered at much greater length in the essay ‘Preserving What? (chapter 6) in this volume.

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When this complaint was considered outside of heated debate, it seemed to come apart in one’s hands. To say that forcing (or any of its fixed-level modal cousins) does not accord with truth-preservation is to say that some inferences licensed by the latter are not licensed by the former. This is without doubt true, but those inferences are precisely the ones that forcing is designed to block. In other words the complaint seems to come down to the fact that forcing doesn’t behave as stupidly as classical inference in the face of (many) inconsistent premise sets. To which one might well reply ‘And . . . ?’ But since merely preserving truth opens the door to a lot of inferences we’d rather not draw, perhaps forcing & Co. can still be thought of as preserving something worthwhile. And it is relatively easy to see that forcing preserves level, which includes preserving truth when the premises taken all together are capable of being true.16 So level preservation can be thought of as a way of fixing truth-preservation at the precise place where it is broken—when there is no truth to preserve. In that case, forcing finds something else, or rather something in addition, that can still be preserved even while truth is on vacation. Now the dialethists can and do argue that they are the ones who really fix truth-preservation, since their notion of paraconsistent reasoning preserves truth and not something other or over and above, like level. Unfortunately this claim stands or falls with the claim that both a sentence and its negation can be true on the same occasion of evaluation. This is a major sticking point, even a deal breaker, for many of those outside the dialethic school. Not to put too fine a point on the matter, an uncharitable person might remark that dialethism, rather than fixing truth-preservation, breaks it beyond any hope of repair.

1.5

The Essays

In the essay ‘Paraconsistency: Who Needs It?’ Ray Jennings and Peter Schotch present an informal account of the core motivating exam16 Which is of course to say that the set of premises is consistent. This is taken up in much more detail in ’On Preserving’ chapter 6in this Volume.

1.5. THE ESSAYS

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ples that lead to a preservationist approach to paraconsistency. They take up the usual examples of belief and obligation and also a not so usual one in recasting the motivation of C.I. Lewis for his notion of strict-implication. In effect, they suggest that Lewis was a pioneer of paraconsistency. In the essay ‘Weakly Additive Algebras and a Completeness Problem’ Alasdair Urquhart presents a solution to the problem of showing that the Kn logics are complete with respect to their intended semantics. This is the problem mentioned above that had to wait several years for its solution. At the same time, Urquhart presents his result as an extension of the original algebraic approach of Tarski and J´onsson, which approach in effect ‘started the whole thing off.’ In the essay ‘Polyadic Modal Logics and Their Monadic Fragments’, Kam Sing Leung and Ray Jennings present the long awaited treatment of n-ary modal logics, and in a particularly elegant way. In ‘Preserving What?’ Gillman Payette and Peter Schotch examine the somewhat shaky status of the truth-preservation paradigm for right reason and suggest alternatives. They treat the concepts of level and forcing with much greater precision and generality than we manage above. In ‘Preserving Logical Structure’ Gillman Payette explores properties of the structure of consequence relations and languages that are preserved in the ‘move up’ to the forcing relation. A number of crucial things about compactness in the context of level are proved, and these results are used to derive a very important theorem concerning the place of forcing among all other similar inference relations. The work on compactness is based on joint work with Blaine d’Entremont. In ‘Representation of Forcing’ Dorian Nicholson and Bryson Brown show how to represent the forcing relation in terms of a more syntactical presentation of the logic. In a traditional context, what we here call representation would be axiomatization. They then extend these results to the more general sequent logic-like case, in which the relation is reconceived as a relation between sets of formulas (premise sets) and sets of formulas (conclusion sets). This important essay corrects a number of mistakes in the earlier ‘On Detonating’ (Schotch and Jennings, 1989).

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‘Forcing and Practical Inference’ is the essay in which Peter Schotch takes up the issue of modifying the definition of forcing in two different ways. In the first modification, the notion of level upon which the forcing relation is based, is required to respect certain natural ‘clumps’ of premises in the sense that no logical cover of a set of premises can break up any of these clumps. The second variation recognizes that logical consistency is a very weak idea and that in ordinary and technical reasoning both, we recognize that the collection of ‘bad guys’ extends far beyond the logically absurd. This recognition requires an obvious change in the definition of logical cover. In ‘Preserving Ambiguity Measures’ Bryson Brown steers preservationism into previously uncharted waters. By defining ambiguity measures as an alternative to partitions for getting something consistent out of an inconsistent set of sentences. Brown not only produces a property worthy of preservation but also a way of connecting preservationism to what was once thought of as an incompatible way of dealing with inconsistency—dialethism. This is rather startling to say the least. The essays are, generally speaking,17 capable of standing alone though there is a considerable overlap. All of them give short weight to some or other important feature (or features) of the preservationist program though none performs such disservice to all the important features. In fine, the essays act together to fix each others’ expository flaws. While the flaws themselves make the essays more readable than completeness would allow.

17 J.

S. Minas used always to add the gloss: ‘which is to say, in no particular case.’

Two

Paraconsistency: Who Needs It? R AY J ENNINGS AND P ETER S CHOTCH

Abstract In this essay Ray Jennings and Peter Schotch present some of the central motivation for what has come to be called the preservationist approach to paraconsistent logic.

2.1

Introduction

The classical account of consistency is not adequate in many situations in which we certainly require a notion like it. This is tantamount to saying that there are situations in which classical logic is not adequate. While such an assertion might have enjoyed a certain amount of shockvalue in the early 20th Century, things have changed since those heady days. Now most people agree that some changes are often necessary in the classical account of inference, though there is nothing approaching wide-spread agreement on the precise nature of those changes. One can distinguish two broad strategies for fixing a misbehaving logic. What we count as misbehavior here is the licensing of inferences which we imagine ought not to be licensed, or the refusal to recognize 15

16

CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

some inferences as valid which we ordinary reasoners find entirely congenial and unproblematic. The strategies referred to just now we may term replacement and revision. When a logic misbehaves in the second way, we most frequently revise it. We don’t need to look far for an example: The logic of ‘unanalysed sentences’ fails to recognize the correctness of the move from ‘All women are mortal’ and ‘Xanthippe is a woman’ to ‘Xanthippe is mortal.’ Annoyed by this denial of such an obvious example of correct reasoning, we call for the logic of sentences to be extended to include terms and individual quantifiers. We are inclined to take as characteristic of the revision strategy that the revised logic contain the one which had revision visited upon it. When a logic misbehaves in the first way, things aren’t quite so clear. At first blush it seems that all we can do is sue for replacement of the offending logic. This is certainly what C.I. Lewis seemed to be doing in the early 20th Century, when he proposed that so-called material implication be replaced with what he called strict implication. Nor was he alone in pointing to perceived flaws in the classical account of reasoning and suggesting that it be given up at least partially, if not root and branch. The purveyors of many-valued logics and the intuitionists to name just two, were also early adopters of the replacement strategy. Now ‘replace’ is a strong word, and we must temper our understanding of it in this context with the realization that the classical account of inference is as inclusive as it could be, barring triviality. If we were to take any inference from propositional logic which is not classically valid, and add it to classical logic as an extra principle, the logic so constituted would allow any formula to be derived from any premise set—which is to say it would be trivial. The upshot of this fact, first proved by Post is that any non-trivial logic over the classical language, is almost sure to be a sublogic of classical logic. So a logic offered for replacement is not normally going to depart from the classical canon in the sense of being disjoint from it. This is certainly the case with intuitionistic logic and many-valued logics, which are easily shown to be included in classical logic. This leaves all those logics which, like the Lewis ones, wish to re-

2.1. INTRODUCTION

17

place the classical  with a connective more like what we really mean when we use words like ‘implication.’ In many, but not all of these cases replacement is the order of the day. But even in these cases, it has sometimes turned out that the proposed replacement is actually a revision, with the new notion of implication representable as some species of modalized classical conditional. This befell the C.I. Lewis program, much to his chagrin one suspects, but we shall revisit his motivation a bit later in this essay to see if any lessons remain to be learned from it. There is an important socio-historical point to be made here. Classical logic is not so-called because of its antiquity;1 it is in fact an invention of the late 19th and early 20th Century. When the first calls for replacement went forth, philosophers and mathematicians were not being asked to jettison some cherished theory handed down by their respected forefathers. Instead, they were being asked to reconsider their allegiance to a relatively new and untried theory. That situation has changed. By the turn of the 21st Century classical logic had become entrenched. By this is meant that the scientific community now has a very large investment in the theory. In practical terms it means that replacement theorists are going to have uphill sledding, and even revisionists will have to endure a certain lack of goodwill if not downright surliness. Of all the replacement theories, one must give pride of place to intuitionistic logic. There is no single reason for this notable success, but one may discern at least two strands to the story. The first is that intuitionism began early enough not to have to fight an entrenched opponent. Trench warfare being what it is, this is a weighty consideration. The second is that what we might call the structures which match the formalized account of the intuitionist calculus2 , seem to arise in many different parts of formal science.3 This is often taken to be an outward and visible sign of the worthiness of the logic displaying it.

1 As

one wag has put it, classical logic is so-called because of its name. what irony that the formalized part of the intuitionistic program carry so much of the weight of promoting the popularity of that program. Intuitionism arose historically as a protest against formalism. 3 Topology, category theory, and quantum physics are three examples. 2 But

18

2.2

CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

The Strange Case of C. I. Lewis

Of all the replacement theories which have been co-opted by classical logic into revision theories, the best known is modal logic. C.I. Lewis lived long enough to see his recommendations concerning strict implication dissolve, by turns, into a proposal to treat the modals as new connectives to be added to the classical store. Of course the proposal even in this new revisionist garb was far from non-controversial. Quine and his students were the most prominent adversaries of this idea and they (and others) were able to cast a pall over modal logic as an area of research. This pall had dispersed by the late 1960’s however and the area has been as respectable as any other since then. To put the early controversy in a nutshell, Lewis thought that the classical account of implication was wrong. By this he meant that some of the classical theorems were wrong when we read the symbol  as ‘implies.’ In defence of this view he would produce what had come to be called ‘paradoxes of material implication.’ Most prominent among these were: ˛  .ˇ  ˛/ :˛  .˛  ˇ/ where these were required to bear the respective interpretations ‘A true proposition is implied by any proposition’ and ’A false proposition implies any proposition.’ According to Lewis, genuine or what he called strict implication must be interpreted in terms of deduction. In his informal semantics of implication then, we would only say that ˛ implies ˇ provided there is some way of deducing the latter from the former. To briefly rehearse some of the more puissant of Quine’s criticisms of this position: Implication is a relation rather than a connective. When we read  as ‘implies’, as Russell often seems to do, this is nothing more substantial than a fac¸on de parler. When we read, for instance ˛  ˇ, as ˛ implies ˇ, we intend nothing more mysterious than the

2.2. THE STRANGE CASE OF C. I. LEWIS

19

assertion that ˛ and ˇ (in that order) are connected by . As for ’real’ or ‘genuine’ implication who knows what that is?4 Actually, as devastating as the criticism might be, Quine could have done rather better than this. Alas, he was one of those who had been so steeped in the 19th Century approach to logic that he took the so-called logical truths to be what logic is about. Once we rid ourselves of that unfortunate affliction, we may say to Lewis: ‘It’s a bit silly to insist that we have a genuine implication (or strict implication if you must) between the propositions ˛ and ˇ only when one can deduce ˇ from ˛. This is silly because we already have a representation for that in our logic, namely `. So your strict implication is simply another word for provability—and why would we need to rename provability?’ Thus do we sound the death-knell for Lewis’ complaint about classical logic. It is simply based on a mistake or, to be as charitable as possible, a confusion between which matters belong to the object language and which to the metalanguage.5 Once we see this, we can dismiss the complaint with its fear-mongering talk of paradox as airily as Quine and his followers. Or can we? As the old saying has it, there is no smoke without fire. Even if Lewis was partially blinded by the smoke, there might yet be a hot coal or two which remains once the smoke has been contemptuously dispersed. To say this is to say that perhaps all this talk of paradox reverberates into the metalanguage. We shall consider this radical idea more carefully. Start with ‘a false proposition implies an arbitrary proposition’ which is impressive enough to have the Latin name ex falso quodlibet. There is an exegesis of this which puts it squarely at the metalinguistic level: ‘If all we know about some premise set is that it contains a falsehood, then that is sufficient to allow the inference of any formula at all as a conclusion.’ Symbolically: f˛; :˛g ` ˇ 4 This is the Quinean sense of not knowing what X is, on which I don’t know what X is, and if you think you do, you’re mistaken. 5 This is understandable for Lewis, who never really accepted the concept of a metalanguage. At least there is no textual evidence that he did.

20

CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

This is no longer about implication, it is now an observation and, dare we say it, a complaint, about the classical account of provability. When Lewis says, as he does of the ‘paradox,’ that there is no genuine sense of deduction in which this assertion makes sense, he echoes generations of beginning logic students who think that it is cheating to use reductio to derive some formula ˇ from a set of assumptions which already contains a contradiction before we assume (for reductio as we blithely say) :ˇ. There is no sense in which the assumption of the hypothesis leads to an absurdity. To assert the contrary is like saying of a man born with a single arm, that the first sight of his mother’s face led to his losing an arm. We must take care here, since Lewis would distinguish the case in which all we know is that a premise is false, from the case in which we know that a premise is necessarily false. For Lewis, the paradigm of a necessarily false premise is a self-contradictory or, as we often say, an absurd one. In this case, says Lewis, bolstered by a famous proof, we really and truly can deduce, using entirely non-controversial rules of proof, anything at all. The dual ‘paradox’ that if all we know is that a conclusion is true, then it may be derived from any premises at all, requires an approach to inference in which we allow sets of conclusions on the right hand side of `. These right-handed sets are understood dually to the left-handed ones. In other words while the comma in f˛1 ; : : : ; ˛n g is understood ‘conjunctively’ if the set is left-handed, it is understood ‘disjunctively’6 if the set is right-handed. So to say that all we know is that a (righthanded) set contains a truth, is to say that the set in question contains something like f˛; :˛g and in that case classical logic tells us that € ` f˛; :˛g for any set of premises €. If we follow Lewis once more we must distinguish this case from the one in which the conclusion set contains a dual-self-contradiction, namely a tautology. Now the fact is, as the complaints of our students indicate, classical 6 The reason for placing these terms in quotes is that we cannot understand a set of formulas disjunctively or conjunctively if the sets in question are infinite. Too, there may be a problem if either of disjunction or conjunction does not occur in the object language.

2.3. THE INCONSISTENCY OF BELIEF

21

logic does not permit us to make any distinction of this sort. And if we think about it for a bit, we may be able to find other reasons to criticize classical logic on that ground.

2.3 The Inconsistency of Belief We begin with an historical example—Hume’s Labyrinth: I find myself in such a labyrinth, that, I must confess, I neither know how to correct my former opinions, nor how to render them consistent. It is no good telling Hume that if his inconsistent opinions were, all of them, true then every sentence would be true. Even if he could accept this startling claim, it would have brought him no comfort. The most which might be wrung out of it is that not all of his opinions could be true. This, so far from being news to Hume, was precisely what occasioned much of the anguish which he evidently felt. What we need in such circumstances is a way to cope. Nor is it merely in the realm of metaphysics that inconsistent opinions charm us. In the physical sciences as well we seem to have inconsistent consequences thrust upon us by equally well confirmed hypotheses. And even if, as we reassure ourselves, the inconsistences are not ineluctable, we must live with them whilst awaiting the crucial experiment or the new paradigm, or the new metaphor, or merely promotion. Eventualities may render the inconsistencies resolvable or illusory or unimportant. In the meantime however, we must draw inferences from them which do not depend upon their inconsistency, but which are informed by it. A more humanistic logic is required. Such a logic would accord with what we must frequently do, namely the best we can with data which, although inconsistent, are nevertheless the best data we are able to command. We would like to be able to reflect but also judge the reasonings of ordinary doxastic agents. We wish to offer them standards of correctness which survive conflicts of belief without triviality. This humane approach need not pander. It would accept classical consistency

22

CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

as an ideal without the pretense that it is pervasive or even especially common in actual belief sets. We say nothing controversial when we claim that often, perhaps usually, the set of sentences to which someone will actually assent would, if scrutinized, be discovered to be classically inconsistent. The question is: What are we to make of this observation? There appear to be two distinct reactions. The first is to say that some such apparently inconsistent belief sets are real and constitute a genuine difficulty for the received theory of inference. The second takes such inconsistencies to be mere appearance—a fanciful mask over the underlying consistent reality. Our choice between these is important, for our ordinary understanding of rationality involves drawing inferences from our beliefs. We are not positively required to draw these inferences but we are required to accept the conclusion once the validity of such an inference has been made known to us. But how is non-trivial reasoning possible in the face of inconsistency? We are like jurors made trusting by cruel penalties. To the extent that we accept the classical account of reasoning, to that extent we resist the view that the set of a person’s beliefs can really be inconsistent in a full-blooded sense. We want to either to acquit or somehow diminish responsibility. We have recourse to such strategies as distinguishing between active beliefs, which must indeed be consistent, and mothballed beliefs which lie strewn about in their cells unregarded and out of mind. We will be inclined to wink at inconsistencies among this disused bricabrac or between them and the brighter ornaments of present thought. It is tempting to think of this distinction as being congruent with the distinction sometimes made between explicit and implicit contradiction. We are happy to offer forgiveness, if not pardon, for the latter but visit upon the former harshness without mitigation. We must step carefully here for the implicit/explicit distinction can be variously applied to confusing effect. We may wish to say that there is an explicit contradiction in a person’s beliefs when she actively adheres to two or more mutually contradictory opinions. Implicit contradictions, on this score, would be those between two inactive beliefs or between one which is active and another which isn’t. This is not Hume’s predicament.

2.3. THE INCONSISTENCY OF BELIEF

23

His problem lay in his active acceptance of contradictory opinions. On the latest account of the distinction, Hume’s contradictions were explicit. Yet he did not recognize whatever harsh penalties later logic would have meted out to him. It did not trivialize his inquires. It simply provided him with ‘a sufficient reason to entertain a diffidence and modesty in all my decisions.’ There is an alternative account of the distinction according to which Hume’s contradictions would count as implicit rather than explicit. On this account, an explicit contradiction is a single sentence which is selfcontradictory; while for one’s set of beliefs to contain an implicit contradiction one must maintain mutually inconsistent opinions, whether actively or inactively. The two ways of drawing the distinction are not always kept separate and the fear of explicit contradictions of the second sort may make us wary of the first sort also. The prohibition against explicit contradictions of the second sort may be expressed as the requirement that every member of a set of beliefs must be logically capable of being true. This stricture seems plausible enough but it does not preclude sets of beliefs which contain implicit contradictions of the second sort. Moreover, it seems likely that implicitly inconsistent belief sets of the second sort are quite common.7 On the other hand the notion of someone believing an explicit contradiction of the second kind seems devoid of content. One might argue that it is part of the root meaning of ‘belief’ that the sentence expressing it must be at least capable of being true. If the classical penalties of triviality are to be imposed anywhere, let them fall upon a transgression which no one is capable of committing. We feel no qualms at the prospect of requiring anybody who believed an explicit contradiction to believe everything. The stronger restriction upon belief sets corresponds to the requirement that such sets be satisfiable, which is to say that all the members be capable of simultaneous truth. It is this condition that flies so directly in the face of our ordinary experience of belief. We have now arrived at a crux. 7 Some

have argued that to be rational one must have such a belief set. See (Campbell, 1980).

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CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

If we let ordinary experience (not to say common sense) be our guide allowing that our belief sets to contain implicit contradiction (of the second kind) but not explicit contradictions, we must of necessity abandon classical logic. For by means of the classical picture of inference we may always derive an explicit contradiction from a set of beliefs which is implicitly inconsistent. Thus classical modes of reasoning trample what is manifestly a useful distinction—the one we earlier attributed to Lewis. How far ought this felt need for a non-classical logic to be indulged? Appeals to experience can easily lead us astray. After all, almost anybody who proposes some species of non-classical logic is tempted to argue that since the world is non-classical so should our logic be. To travel too far along this road is to invite some form of the genetic fallacy. If our arithmetic or our syllogisms are not in accord with the classical canons, the fault might well lie in our inability to calculate or to construct Venn diagrams. The observation that the reasonings of actual people are sometimes non-classical, in order to bite, must be coupled with evidence that such reasoning is correct, or at the very least not evidently mistaken. If invention is to be mothered let it be through necessity and not through the failure to take suitable precautions. Nevertheless some of the evidence is compelling. Even when our stock of beliefs is inconsistent we routinely draw a distinction between what follows from it and what does not, and regard certain inferences from our beliefs as improper, in spite of the circumstances. If our procedures were genuinely classical we would not, indeed could not, make such a distinction. There is no classical issue to be taken with any inference from an inconsistent set. Failure to take this sort of evidence into account seems as misguided as any of the worst excesses of those who would base logical theory democratically upon the actual inferential practice of the proletariat. It is no doubt true that if the rule If A then B; B; therefore A. (sometimes known as Modus Morons) were from time to time invoked

2.3. THE INCONSISTENCY OF BELIEF

25

in country districts or local government, we should nevertheless resist viewing it as a ratiocinative dialect, or alternative inferential lifestyle. But if humanity universally manages to distinguish, at least in some cases, valid arguments from inconsistent premises on the one hand from invalid arguments from those same premises, this is another and weightier matter. The view that everyone commits a logical error in making such a distinction is an adaptation for logic of the doctrine of original sin. We have already considered one non-classical view of the matter, namely the view that when we argue from our beliefs we disregard some, reasoning only from the active portion of our belief set. That this is non-classical is evident from the fact that it is non-monotonic. Whatever sense of inferable we adopt, it is clear that we do not think of our conclusions as inferable in that significant way from an inconsistent set. The reason for such a non-monotonic stance is precisely that the whole of one’s belief might be seen to be inconsistent if examined, that an inference drawn from the set taken as a whole might be trivial. To escape this consequence we must take the further step of claiming that the only real beliefs are the active ones. This protestant view, which is really an attempt to deny that beliefs can be inconsistent, suffers from two flaws. The first is that on this doctrine it is nigh on impossible to identify a belief set. For suppose someone’s apparent belief set is inconsistent. Which subset of the set of apparent beliefs constitutes the real belief set? (assuming that the real beliefs are also apparent; otherwise the problem becomes even worse). Even if we insist that the real beliefs form a maximal consistent subset of the apparent beliefs we do not thereby guarantee uniqueness. We must elaborate more sophisticated side conditions to ensure this, perhaps invoking the probability calculus to assist. In this case we must have handy some interpretation of probability other than one of its belief interpretations. The project seems a mare’s nest. But these difficulties would not deter us from the task were it not for the second flaw. The second flaw is that the basic premise is false. Try any thinking beyond the wallpaper or musak variety of daily life, and inconsistent pairs of beliefs do turn up, the one belief as active as the other.

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2.4

CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

Inconsistency and Ethics

It seems to be part of our ordinary moral experience that we can find ourselves in a dilemma, which is to say a situation in which the demands of morality cannot (all) be heeded. Let us suppose that such situations might extend beyond the world of everyday all the way into philosophy.8 Let us next ask the question of whether or not obligations9 are ‘closed under logical consequence.’ This is an example of a question which is both of interest to ethical theory, and also to logic. Some philosophers, even some ethicists, might object that the question is one that only a logician could love, and that they, as ethicists, have not the slightest interest in it. Such a position betrays a lack of thought. The issue is really quite deep. If you deny that the logical consequences of obligations are also obligations, then you are saying that moral argument, at least moral argument of a certain standard kind, is impossible. When you and I enter into a moral dispute, a dispute over whether or not you ought to bring it about that P say, then what often, perhaps usually, happens is that I try to demonstrate that P follows from something, some general moral principle, to which you subscribe. In stage one then, we obtain the subscription. Don’t you think that we ought to help those who cannot help themselves? Not in every case, suppose somebody is both helpless and not in need of help? Very well, we ought to help those in need who cannot help themselves? Wonderfully high minded, but impossible I’m afraid. Why impossible? We are overwhelmed by the numbers don’t you see? The numbers of what? Those in need? 8 Such a supposition is by no means beyond the bounds of controversy. There are moral philosophers who argue that such conflicts of obligation are merely apparent. 9 We are sacrificing rigor in the cause of clarity in this example. Strictly speaking we are not talking about ‘obligations’ at all, since those are typically actions. We should be talking rather about ‘sentences which ought to be true.’ This is a much more awkward form of words however and it would interfere with the flow without adding very much to what we are saying.

2.4. INCONSISTENCY AND ETHICS

27

Precisely! We would spend all our resources helping the needy only to join their ranks before long. Ah, well how about this modification then: We ought to help those in need who cannot help themselves, but not to the point of harming ourselves or even seriously inconveniencing ourselves? Yes, it doesn’t sound so high-minded, but now it’s something I can support. Now we close in for the kill. Would you say that Jones is in need of help? Yes, not much doubt about that, poor bloke he. We should pass the hat, don’t you think? Everybody could put in what they can easily spare. I don’t see the point in that. Do you imagine then that Jones can pull himself up by the bootstraps, that all he needs is sufficient willpower? No, I wouldn’t say that, not in his present state at least. Then, we ought to help him, wouldn’t you say? Well somebody needs to help him if he’s going to be helped, but I don’t see how that got to be my problem. What has been shown is that ‘We help Jones’ is a logical consequence of ‘We help those in need who cannot help themselves and whose help is not a significant burden to us.’ This much is non-controversial. Having first agreed upon ‘We ought to help those in need ...’. But the remaining step, the step from the forgoing to ‘We ought to help Jones’ will only follow if the logical consequences of obligations are themselves obligations. It is just crazy to say that the issue is of no interest to ethicists, unless the ethicists in question have no interest in arguments like our sample. Nor can the relevance of logic to ethics be exhausted by this one issue. Suppose that in exasperation, our stubborn ethicist agrees that the issue is of interest after all. But the interest is over once we see that we must answer ‘yes’ to the question of whether or not the consequences of

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CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

obligations are themselves obligations, isn’t it? Not at all. Recall how this got started. We were wondering about moral dilemma, about conflict of obligations. In view of what we just said about obligations and logical consequences, we now have a problem. Suppose I am in a moral dilemma, that there are two sentences P and Q which I believe should both be true (I believe that each, individually, ought to be true). But the two cannot be true together—hence the dilemma. This is bad enough, but it gets worse. One of the consequences of the set of sentences which we say ought to be the case, the set of obligations, is the sentence P ^ Q. But since the two sentences cannot both be true at the same time, this consequence is equivalent to P ^:P . Since the consequences are also obligations, P ^ :P is an obligation. But everything follows from P ^ :P so in the case of a moral dilemma, one is obliged to do everything. This seems a bit burdensome.

2.5

Summing Up

The unhappy fact of the matter is that sometimes, through no fault of our own, we are simply stuck with bad data. This can happen in any of the variety of circumstances in which we gather information from several sources. Should the sources contradict each other, we may have a way of measuring their relative reliability in some reasonable way. Equally often however either the means at our disposal will not settle the matter conclusively (as in the case of beliefs) or we simply have no means for the adjudication of conflict between our sources. The classical theory of inconsistency must retire in such a situation, since non-trivial classical reasoning is impossible in the face of inconsistent premise sets. All that the classical logician can advise in these circumstances is to start over again with a consistent set of premises. Sometimes this is wise counsel but more often it is of a piece with what the doctor said when told by a patient: ‘It hurts when I do this.’ To get back to our earlier example, both belief and obligation seem to call for the Lewis distinction (at least the one that remains after we ‘fix’ the original claim). We need to be able to distinguish premise

2.5. SUMMING UP

29

sets which contain full-blooded self-inconsistencies from those which do contain pairs of premises which are inconsistent although the individual premises are not. This is the path which we shall follow—the one we shall eventually call preservationist. But we should point out that there is another path which is quite distinct from ours but which claims for itself all the merits of our approach and perhaps others besides. This other account may be distinguished from ours by being a wholehearted replacement theory, where ours is a revision theory. According to the account dialethism, as its defenders have come to call it, there can be premise sets in which both a sentence A and its negation not-A appear without any ill effect. Without that is, the premises having as the set of consequences, the set of all sentences. What is more, such a set will also contain the single conjunction ‘A and not-A’. Subscribers to this account suppose that the true meaning of ‘not’ is such as to make ‘A and not-A’ capable of being true. This move allows the satisfiability condition to reappear since many more sets (perhaps even every set) will be satisfiable. As a replacement theory this has all the drawbacks of such when the target of replacement is classical logic. But quite apart from the resistance of vested interests, the dialethists must also endure the scorn of even the naive and uncommitted. The very idea of ‘true contradiction’ seems to be about as counterintuitive as one could imagine, unless one is addicted to continental philosophy of the more literary sort. In mincing negation, this scheme avoids the classical consequences of contradiction by changing the meaning of ‘contradiction.’ Let us say at once that the dialethists have many virtues. The boldness of their approach is admirable, and they have been far from lazy. Both the theory underlying the move and its inferential consequences have been energetically worked out and widely published. But where one ought to feel release and fill one’s lungs, one feels a kind of dull unease. For even if classical negation is not primordial in human language, but merely a contrivance of latter days, it is nonetheless with us and even if not all, at least some of our contradictions seem to be on that scale.

Three

Weakly Additive Algebras and a Completeness Problem A LASDAIR U RQUHART

Abstract In this essay an extension of existing approaches to the algebra of modal logic is studied. Where the originating work by J´onnson and Tarski concerned the representation of Boolean algebras with additive operators, the present effort considers the representation of bounded distributive lattices with weakly additive operators. Where J´onsson and Tarski rely upon Stone’s representation theory for Boolean algebras, this relies upon Priestley’s representation theory (usually called Priestley duality) for bounded distributive lattices.

3.1

Introduction and Brief History

In a series of papers,1 Ray Jennings and Peter Schotch have developed a generalized relational frame theory that goes beyond the standard approach in modal logic Schotch and Jennings (1980a,d); Jennings and 1 For further details concerning the origin of these papers see the the introductory chapter of this volume.

31

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CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

Schotch (1981a, 1984). The key idea is to generalize the usual truth condition for the necessity operator by using multi-place relations rather than the usual binary relation. Thus, if R is an n C 1-place relation defined on a set W , then Jennings and Schotch state the truth condition for the generalized necessity operator as follows. If x 2 W , then x ˆ ˛ ” 8y1 ; : : : ; yn ŒRxy1 ; : : : ; yn ) 9i.yi ˆ ˛/: This definition validates the scheme Kn of n-ary aggregation: _ ˛0 ^


^ ˛n  Œ ˛i ^ ˛j : 0i <j n

In Jennings and Schotch (1984), the authors claim without proof that the class of formulas valid in all frames based on an n C 1-place relation is axiomatized by the scheme Kn , together with rules of necessitation and monotonicity. This claim is proved by Apostoli and Brown (1995a). An earlier completeness proof was given in Schotch and Jennings (1980a), which exploited that idea that necessity operators and inference relations are related in a certain way which is reminiscent of a Galois connection. As clever (and fecund) as this latter approach is, it is not in the mainstream of modal logic. It isn’t, in other words, pure since it brings in ideas that are not part of what we might call classical modal logic. This is because it makes essential use of a non-classical inference relation (called n-forcing). The same holds for the proof by Apostoli and Brown since it depends on an intermediate lemma stating chromatic compactness for hypergraphs. Though the Apostoli Brown proof was substantially simplified by Nicholson, Jennings and Sarenac (Nicholson et al., 2000b), the notion of chromatic compactness was still used. There is now a proof due to Dorian Nicholson , detailed in chapter 4, which is probably the best one can do in the way of a pure proof. A proof of completeness for the logic Kn is presented in 3.4 of this essay, where it is derived as a corollary to the main representation theorem for weakly additive algebras. Is this also pure? It certainly brings in something that seems external to classical modal logic, but how much is in that seeming? One might well argue that the representation theory of certain sorts of algebras is the basis of classical modal semantics,

3.1. INTRODUCTION AND BRIEF HISTORY

33

whether or not modal logicians acknowledge the fact. Agree to that, and you agree that the proof below is pure. In the early 1950’s Bjarni J´onsson and his thesis advisor2 Alfred Tarski began work on the representation of Boolean algebras with extra operators. One is tempted to conjecture that Tarski was the one who suggested this topic which was a kind of melding of his friend Marshall Stone’s ground-breaking work on the representation of Boolean algebras3 , and his own work with J.C.C. McKinsey on the algebra of closure operators in, for example, (McKinsey and Tarski, 1948). Their joint work was published in (J´onsson and Tarski, 1951, 1952). The representations in question work like this: We can easily see that every finite Boolean algebra is ‘the same’ as an algebra of sets, namely the set of all subsets of some finite set, called the underlying set, with the Boolean operations of meet, join, complement, 1, and 0 represented by set intersection, set union, relative set complement, the underlying set and the empty set. We can spell out the sameness in several different (and equivalent) ways. One of them is: There is a translation T which maps the atoms of the Boolean algebra (all finite Boolean algebras are atomic) to the unit subsets of the underlying set of the set algebra and the equation x ^ y D 1 holds in a given Boolean algebra, with underlying set † if and only if T .x/ \ T .y/ D †. The algebraic point to make here, is that such a translation T can be reversed by the obvious inverse translation T 1 which takes unit sets to atoms, intersections to meets etc. Moreover, the existence of these two translations amounts to an isomorphism between the Boolean algebra and the set algebra. This is pretty much old news by the 1930’s. But it raises a question: can this result extend to arbitrary Boolean algebras? In other words is it the case that every Boolean algebra whatever is isomorphic to an algebra of sets? There is a lot of content in the ‘whatever,’ since there are infinite Boolean algebras and, also non-atomic ones. Stone proved that this question has a positive answer though the set algebra in question 2 In fact,

J´onsson was the very first graduate student to work with Tarski after the latter’s move to the United States. 3 Which work is reported in Stone (1936).

34

CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

isn’t what the novice might expect. The sets, in general, are drawn from a topological space and have certain curious properties. They are clopen, which is to say both both open and closed. J´onsson and Tarski consider the problem of what changes must occur to preserve this isomorphism when we introduce extra operations. They consider the case of a single n-ary operation, which we shall call Þn (with the subscript suppressed when no confusion will result) which is a generalization of the closure operation. What they show is that such expanded Boolean operations can be represented by the original set algebras provided that we introduce an n+1-ary relation to deal with the added operator. The translation which makes everything work is that the generalized closure Þn .x1 ; : : : ; xn / goes to the set of sets fb1 ; : : : ; bn g such that a is an element of T .xi / and Rab1 : : : bn . Example 1. The ordinary closure is a unary operation. In that case the closure Þ.x/ is represented by the set of elements b such that a 2 T .x/ & Rab. It is easy to see the connection between this condition and the usual ‘Kripke’ truth condition for possibility.4 Of course not any old generalized closure operator will be representable in this way. In fact, J´onsson and Tarski restrict their operators to those they call additive. Their additivity condition was expressed by the Boolean equality: Þn .x1 ; : : : ; xi _ yi ; xiC1 ; : : : ; xn / D Þn .x1 ; : : : ; xi ; xiC1 ; : : : ; xn / _ Þn .x1 ; : : : ; xi

1 ; yi ; xi C1 ; : : : ; xn /;

In addition to additivity, the normality condition which is to say that Þ.x1 ; : : : ; xi ; 0; xiC1 ; : : : ; xn / D 0, was also usually imposed on the generalized closure operators. Sometimes conditions are expressed by means of inequalities in order to stay closer to logical formulas which are most often expressed by conditionals. If the above condition were so-expressed the condition (upward) monotonicity, which is to say that Þn .x1 ; : : : ; xn /  4 In his Kripke (1963) Kripke refers (in a footnote) to the work of J´ onsson and Tarski more than a decade before his publication as a surprising anticipation, though it isn’t clear to the casual reader who should be (most) surprised.

3.2. WEAKLY ADDITIVE OPERATORS

35

Þn .y1 ; : : : ; yn / whenever xi  yi for 1  i  n, would have to be added in order to express additivity. This condition applies in particular to the algebras which correspond to the Jennings and Schotch Kn logics in which the modal formula is a conditional. The present essay sets the modal-logical results in a broader framework, by presenting the theory of Jennings and Schotch as a generalization of the work of J´onsson and Tarski. To be consistent with this earlier work, the theory is couched in terms of the possibility operator rather than the dual necessity operator used in the work of Jennings and Schotch (and in modal logic generally speaking). The emphasis on possibility does not reflect some preference for this modality over necessity, but rather a desire to focus on the notion of closure which is more likely to be familiar to mathematicians. The topological equivalent of a necessity operator is an interior operator which, while not unfamiliar, is not as popular—topologically speaking.

3.2

Weakly additive operators

Let L be a bounded distributive lattice. Such an object is a generalization of the (perhaps) more familiar Boolean algebra in the sense that every Boolean algebra is a bounded distributive lattice, but the converse is not true. To take a single (but prominent) example, a Heyting algebra is a bounded distributive lattice which is not a Boolean algebra. E : : : to refer to sequences of eleWe shall use vector notation aE ; b; ments from L, including the zero vector ƒ. Operations and relations on such vectors are to be understood in terms of the product lattices Lk . In other words, a binary operation, let’s call it ^, on two vectors aE and bE is to be understood as resulting in a unique vector aE ^ bE in which the elements of this sequence are defined as the ^ of the corresponding eleE 5 Of course this assumes that the two vectors have the ments of aE and b. same length. We use jE aj for the length of the vector aE ; the vector of length k in which all of the entries are the element a is written ak (when k D 0, 5 This

is often described as defining the ^ of two vectors componentwise.

CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

36

this is the empty vector ƒ). If kE is a sequence of integers, then we write PE k for the sum of the elements in the sequence. In the succeeding section on duality theory, we need notations and definitions for sequences of elements from an arbitrary set. We write x 2 yE if x is a member of the sequence y. E If xE is a sequence of elements from a set S and A  S, then we write xE  A if every element in the sequence xE is in A. T If xE is a sequence of subsets of a fixed set S, then we use the notation xE for the intersection of all T the sets in the sequence x; E in the case of the empty sequence ƒ of sets, ƒ D S. Definition 1. If xE and yE are sequences of elements from a set S with an ordering relation  defined on it, and jxj E  jyj, E then we write xE  yE if xi  yi for all i, where 1  i  jxj: E Let Þ be an n-place operation defined on L. Definition 2. For k > 0, we say that Þ is k-additive in the ith place if it is normal and satisfies the conditions ^ _ ÞŒ aE ; .bh _ bj /; cE   ÞŒ aE ; bj ; cE  0h<j k

for aE 2 Li

1,

cE 2 Ln

i

0j k

and b0 ; : : : ; bk 2 L.

Example 2. Suppose Þ is binary. Then Þ is 1-additive in the second place if and only if Þ.x; .b0 _ b1 //  Œ.Þ.x; b0 / _ Þ.x; b1 / While Þ is 2-additive in the first place if and only if Þ..b0 _ b1 / ^ .b0 _ b2 / ^ .b1 _ b2 /; y/  ŒÞ.b0 ; y/ _ Þ.b1 ; y/ _ Þ.b2 ; y/

3.2. WEAKLY ADDITIVE OPERATORS

37

In the following definition we define something a bit more general than our earlier notion of monotonicity, which applies only where the relation  assumes that the related vectors are of equal length. Definition 3. Þ is said to be monotone if it satisfies the condition: E ÞE a  ÞbE whenever aE  b.

Definition 4. An operator on a bounded distributive lattice is defined to be weakly additive if it is monotone and in all of its argument places it is k-additive for some k > 0.

Definition 5. An additive operator in the sense of J´onsson and Tarski (J´onsson and Tarski, 1951, 1952) is a monotone operator that is 1-additive in all of its argument places.

Definition 6. We say that an n-place weakly additive operator Þ is of type kE if kE is a vector of positive integers of length n so that Þ is ki -additive in its ith place. For example: Example 3. If Þ.a; b; c/ is 1-additive in its first place, 3-additive in its second place and 2-additive in its third, then it is of type h1; 3; 2i. Definition 7. We define a weakly additive algebra of type kE to be a bounded distributive lattice L together with a weakly additive operator of type kE defined on L. If we translate the Kn modal logics considered by Jennings and Schotch into algebraic form, the result is a special case of the general

CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

38

framework outlined above. Consider a propositional logic satisfying the scheme Kn of n-ary aggregation, as well as the rules of necessitation and monotonicity. We can define a possibility operator in the usual way as the dual of the necessity operator; that is to say, Þ˛ $ ::˛. Then the scheme Kn is provably equivalent to the dual scheme: ^ ÞŒ ˛i _ ˛j   Þ˛0 _ : : : _ Þ˛n : 0i<j n

Hence, the algebraic version of the Jennings/Schotch system with the axiom scheme Kn is the theory of Boolean algebras with a one place monotone normal operator Þ satisfying ^ _ ÞŒ .bi _ bj /  Þbi 0i <j n

0i n

that is to say, Boolean algebras with a one place weakly additive operator of type hni. In what follows, we shall call these Jennings/Schotch algebras; we shall denote the variety of weakly additive algebras satisfying the above inequality by the notation Vn . The Jennings/Schotch truth condition for , when translated into its dual form, is as follows: x ˆ Þ˛ ” 9y1 ; : : : ; yn ŒRxy1 ; : : : ; yn ^ 8i.yi ˆ ˛/:

3.3

A Further Generalization

Weakly additive operators can be defined from multi-place relations, in a way that generalizes both the Jennings/Schotch truth condition for possibility, and a construction of J´onsson and Tarski (J´onsson and Tarski, 1951, Definition 3.2). Let kE be a length n sequence of nonP negative integers, X a non-empty set and R a . kE C 1/-place relation on X. We define an n-place operation ÞR on P .X / as follows: Definition 8.

x 2 ÞR .A1 ; : : : ; An / ”

9yE1 ; : : : ; yEn ŒRx yE1 ; : : : ; yEn ^ 8i Œ1  i  n ) yEi  Ai 

3.4. DUALITY THEORY FOR WEAKLY ADDITIVE OPERATORS

39

where for each i, 1  i  n, yEi is a vector of length ki . Theorem 4. For kE a sequence of non-negative integers of length n, and P R a . kE C 1/-place relation on a non-empty set X, ÞR is a weakly additive operator of type kE on P .X /. Proof. Let L be P .X /. We need to prove for k D ki > 0 the inclusion \ [ E E Bj ; CE ; ÞR Œ A; .Bh [ Bj /; CE   ÞR Œ A; 0h<j k

where AE 2 Li

1,

CE 2 Ln

0j k i

and B0 ; : : : ; Bk 2 L. Assume \ E x 2 ÞR Œ A; .Bh [ Bj /; CE ; 0h<j k

so that there exist yE1 ; : : : ; yEn satisfying the conditions: Rx yE1 ; : : : ; yEn ^ 8j .1  j < i ) yEj  Aj / ^8j .i < j  n ) yEj  Cj /; \ yEi  .Bh [ Bj /: 0h<j k

Let us suppose that for all j , 0  j  k, yEi 6 Bj . Then for any such j , there is a z 2 yEi so that z … Bj . Since jyEi j D k, it follows (by the pigeonhole principle) that there are distinct h; j so that for some z 2 yi , z … Bh [ Bj , contrary to assumption. Hence for some j , yEi  Bj , proving the inclusion. The remaining conditions defining a weakly additive operator of type kE are easily verified. 

3.4

Duality theory for weakly additive operators

The duality for weakly additive operators described in this section is piggy-backed on Priestley’s duality for bounded distributive lattices. We begin by giving the main results of this duality theory without proofs. For detailed discussion of the theory, the reader is referred to articles by Priestley (Priestley, 1970, 1984), Davey and Duffus (Davey and Duffus,

40

CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

1982) and to the textbook by Davey and Priestley (Davey and Priestley, 1990). An ordered topological space is a topological space with a partial order relation defined on it. A subset E of a partially ordered set is increasing if x 2 E, x  y imply y 2 E. A map f between partially ordered sets is increasing if x  y implies f .x/  f .y/, and decreasing if x  y implies f .y/  f .x/. A map between two ordered spaces is said to be an order-homeomorphism if it is both a homeomorphism and an isomorphism with respect to the orderings on the spaces. A Priestley space is an ordered topological space S that is compact and totally order-disconnected, that is, for points x; y of S, if x 6 y then there is a clopen increasing set U such that x 2 U; y … U . If L is a bounded distributive lattice, then the dual space of L, S.L/, is the ordered topological space in which the set of points S is the family of all prime filters of L, ordered by containment, and the topology is defined by taking as a sub-base the family of all sets fr 2 S W a 2 rg and fr 2 S W a … rg for a 2 L. Conversely, if S is a Priestley space then the dual lattice of S, L.S/, is the set of all clopen increasing sets of S, with the lattice operations of set intersection and union. Theorem 5. 1. If L is a bounded distributive lattice, then S.L/ is a Priestley space, and L is isomorphic to L.S.L// under the mapping .a/ D fx 2 S.L/ W a 2 xgI 2. If S is a Priestley space, and L.S/ its dual lattice, S is orderhomeomorphic to S.L.S// under the mapping .x/ D fB 2 L.S/ W x 2 Bg: We obtain the duality theory for weakly additive operators by building on Priestley’s duality theory. Let kE be a sequence of non-negative PE E We define a reintegers, and k D n the sum of the entries in k. lational space of type kE to be a structure hS; Ri, where S is a Priestley P space, and R is a . kE C1/-place relation on S , satisfying the following conditions:

3.4. DUALITY THEORY FOR WEAKLY ADDITIVE OPERATORS

41

1. If A1 ; : : : ; An 2 L.S/, then ÞR .A1 ; : : : ; An / is clopen; 2. If Rx yE and x  z then Rz y; E 3. If for all i, 1  i  n, jyi j D ki and :Rx yE1 ; : : : ; yEn then: .9A1 ; : : : ; An 2 L.S// Œ 8i .1  i  n ) yEi  Ai / ^ x … ÞR .A1 ; : : : ; An /. E then the dual algebra If R D hS; Ri is a relational space of type k, of R is the algebra A.R/ defined on the lattice L.S/ by adding the operation ÞR . If hL; Þi is a weakly additive algebra, where Þ is an n-place operator, and each yE1 ; : : : ; yEn is a sequence of filters in L, then the relation RÞ is defined by: \ RÞ x yE1 ; : : : ; yEn , 8E a 2 Ln Œ 8i.ai 2 yEi / ) Þ.a1 ; : : : ; an / 2 x: E then the dual If A D hL; Þi is a weakly additive algebra of type k, space of A, R.A/, is defined by adding to the Priestley space of L the relation R defined by: Rx yE1 ; : : : ; yEn , RÞ x yE1 ; : : : ; yEn ^ 8i Œ1  i  n ) jyEi j D ki ; E where n D jkj. If L is a distributive lattice, then we employ the following terminology for vectors uE , where each ui is a subset of L. For such vectors uE and vE, where jE uj  jE v j, we write uE  vE if for all i where 1  i  jE uj, ui  vi (this terminology is consistent with the corresponding terminology introduced above for vectors of points in ordered sets). If F is a family of such vectors, and the maximum length of a vecS tor in S F is k, then we define F to be the vector vE of length k where vi D fui jE u 2 F g. For a lattice L, x a filter in L, and a an element of L, we denote by xŒa/ the smallest filter containing x and a. The principal filter containing a is denoted by Œa/. E where Lemma 1. Let hL; Þi be a weakly additive algebra of type k, n E jkj D n. Then for aE 2 L , x 2 S.L/,

CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

42

ÞE a2x) 9wE1 ; : : : ; wEn Œ Rx wE1 ; : : : ; wEn ^ 8iŒ 1  i  n ) ai 2

\

wEi  :

Proof. Let ÞE a 2 x, where x 2 S.L/. Define F to be the family : ; uEn , where each uEi is a sequence of filters in of all sequences uE1 ; : : T L, so that: (1) ai 2 uEi ; (2) For every i, 1  i  n, there are b1 ; : : : ; bi 1 ; bi C1 ; : : : ; bn so that for all y 2 uEi , there is an element ay 2 L such that 8z 2 uEi .z ¤ y ) ay 2 z/ ^ Þ.b1 ; : : : ; bi

1 ; ay ; biC1 ; : : : ; bn /

… xI

(3) RÞ x uE1 ; : : : ; uEn ; (4) juEi j  ki . We define an ordering on F as follows. For uE1 ; : : : ; uEn 2 F and vE1 ; : : : ; vEn 2 F , uE1 ; : : : ; uEn  vE1 ; : : : ; vEn if and only if for all i, uEi  vEi . First, we show that F ¤ ;. For ki > 0, set uEi D hŒai /i. Then uE1 ; : : : ; uEn 2 F , since for y 2 uEi , we can set bj D 1 for j ¤ i, and ay D 0, since Þ.1; : : : ; 1; 0; 1; : : : ; 1/ D 0 … x. Second, we show that every chain in F has an upper S bound in F . Let C  F be a chain; for 1  i  n define uEi D fvEi jvE1 ; : : : ; vEn 2 Cg: Then it is straightforward to check that uE1 ; : : : ; uEn is in F , and is an upper bound for C. By Zorn’s Lemma, F contains a maximal element wE1 ; : : : ; wEn . To complete the proof of the Lemma, we need to show that all the filters in wE1 ; : : : ; wEn are prime. Suppose that this is not so, so that for some wEi D y1 ; : : : ; yj ; : : : ; yl , yj is not prime; thus there are d; e 2 L where d _ e 2 yj , but d; e … yj . Let uE ; vE be the vectors that result from uEi by replacing yj by yj Œd / and yj Œe/ respectively. Since wE1 ; : : : ; wEn is maximal in F , it follows that :RÞ x wE1 ; : : : ; wiE 1 ; uE ; wiEC1 ; : : : ; wEn and :RÞ x wE1 ; : :T : ; wiE 1 ; vE; wiC1 E T ; : : : ; wEn . ThusT there are f1 ; : : : ; fn 2 L so that f1 2 wE1 ; : : : ; fi 2 uET ; : : : ; fn 2 wET /…x n , Þ.f1 ; : : : ; fn T and g1 ; : : : ; gn 2 L so that g1 2 wE1 ; : : : ; gi 2 vE; : : : ; gn 2 wEn , and Þ.g1 ; : : : ; gn / … x. Let zE be the vector hE u; yj Œe/i. We claim that RÞ x wE1 ; : : : ;T zE; : : : ; wEn . For ifTnot, there are T elements p1 ; : : : ; pn 2 L so that p1 2 wE1 ; : : : ; pi 2 zE; : : : ; pn 2 wEn and Þ.p1 ; : : : ; pn / … x. But since pi 2 yj Œd / and pi 2 yj Œe/, it follows that pi 2 yj , contradicting RÞ x wE1 ; : : : ; wEn .

3.4. DUALITY THEORY FOR WEAKLY ADDITIVE OPERATORS

43

We now show that the sequence wE1 ; : : : ; zE; : : : ; wEn also satisfies the second of the three conditions defining the elements of F . Let b1 ; : : : ; bi 1 ; biC1 ; : : : ; bn be the sequence of elements associated with wEi by condition (2). For j ¤ i, let hj D bj ^ fj ^ gj . We claim that the sequence h1 ; : : : ; hi 1 ; hiC1 ; : : : ; hn satisfies condition (2). For elements y in zE other than the filters yj Œd / or yj Œe/, we can set ay to be the element ay associated with wE1 ; : : : ; wEn by condition (2). For y D yj Œd /, set ay D gi , and for y D yj Œe/, set ay D fi . Then it follows from our assumptions that h1 ; : : : ; hi 1 ; hi C1 ; : : : ; hn and the elements ay satisfy condition (2). Since wE1 ; : : : ; wEn  wE1 ; : : : ; zE; : : : ; wEn , and wE1 ; : : : ; wEn is a maximal element in F , wE1 ; : : : ; zE; : : : ; wEn … F . Because wE1 ; : : : ; zE; : : : ; wEn satisfies conditions (1) to (3) in the definition of F , it follows that l D jwEi j D ki . Let a0 ; : : : ; al be the sequence of elements ay for y 2 zE. Then ^ \ ai _ aj 2 zE; 0i<j l

so that Þ.h1 ; : : : ; hi

1;

^

ai _ aj ; hiC1 ; : : : ; hn / 2 x:

0i<j l

It follows that

_

Þ.h1 ; : : : ; ai ; : : : ; hn / 2 x:

0il

But then since x is a prime filter, it follows that for some i, 0  i  l, Þ.h1 ; : : : ; ai ; : : : ; hn / 2 x, contrary to the definition of ai . This contradiction completes the proof that all the filters in wE1 ; : : : ; wEn are prime. In the definition of F , we have only required that juEi j  ki , for 1  i  n, while the definition of R above requires the condition juEi j D ki . However, if RÞ x uE1 ; : : : ; uEi ; : : : ; uEn , then RÞ x uE1 ; : : : ; huEi ; zi; : : : ; uEn , where z is any element in uEi . Consequently, we can conclude that \ 9wE1 ; : : : ; wEn Œ Rx wE1 ; : : : ; wEn ^ 8iŒ 1  i  n ) ai 2 wEi  : This completes the proof of the lemma.



44

CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

E then Theorem 6. 1. If A is a weakly additive algebra of type k, E R.A/ is a relational space of type k and A is isomorphic to A.R.A// under the mapping .a/ D fx 2 R.A/ W a 2 xgI E and A.R/ its dual algebra, R 2. If R is a relational space of type k, is order-homeomorphic to R.A.R// under the mapping .x/ D fB 2 A.R/ W x 2 Bg: Proof. (1) The main condition to be verified is that ÞE a2x) 9wE1 ; : : : ; wEn Œ RÞ x wE1 ; : : : ; wEn ^ 8i Œ 1  i  n ) ai 2

\

wEi  ;

for x 2 S.L/. This follows immediately from the definition of RÞ and Lemma 1. The conditions defining a relational space of type kE are then easily verified from the definitions, using this equivalence. Priestley’s representation theorem (theorem 5) implies that  is a lattice-isomorphism, so it suffices to show that  is an isomorphism with respect to Þ. That  is also an isomorphism with respect to Þ is exactly the content of the above condition. (2) By theorem 5, the map  is an order-homeomorphism from R onto the second dual of R, so it suffices to prove that  is an isomorphism with respect to the relation R. This follows from the definition of E ÞR and the third condition defining a relational space of type k.  The representation theorem we have just proved is an algebraic version of the completeness theorem described in the introductory section. In fact, that completeness theorem can be derived from the representation theorem. (theorem 6). Theorem 7. The logic Kn is complete with respect to the family of all n C 1-ary relational frames, given the Jennings/Schotch truth condition for the  operator.

3.4. DUALITY THEORY FOR WEAKLY ADDITIVE OPERATORS

45

Proof. We can construct a Boolean algebra with operator Þ by the familiar Lindenbaum construction; that is to say, we construct an algebra in which the elements are classes of provably equivalent formulas. If we begin from the logic Kn , then the result is an algebra A in the variety Vn . By theorem 6, this algebra is isomorphic to A.R.A// under the mapping .a/ D fx 2 R.A/ W a 2 xg: Consider the .n C 1/-ary relational frame defined on the family of all prime filters in A, with the accessibility relation R defined in the dual space R.A//. For a propositional variable P , the set of points at which it is true is defined to be .ŒP /, where ŒP  is the set of formulas equivalent to P in Kn . It is a straightforward exercise to check that a formula is true at all points in this model if and only if it is a theorem of Kn . 

Four

A Dualization of Neighborhood Structures D ORIAN N ICHOLSON

Abstract A dualization of neighborhood semantics is introduced for the purpose of obtaining a simplified proof that weakly aggregative modal logic is complete with respect to .n C 1/-ary relational frames. This new class of quasi-semantic structures exploits the theory of transverse hypergraphs. Unlike the other proofs in the literature, the one included here does not cite chromatic, or colouring, compactness. Along the way we prove completeness for a denumerable class of non-normal modal logics, which have deontic, as well as philosophical logical, motivations.

4.1

Introduction

The search for a completeness proof for Jennings and Schotch’s weakly aggregative modal logic lasted for nearly twenty years (cf. Johnston (1978); Schotch and Jennings (1980b); Jennings and Schotch (1981b); Schotch and Jennings (1980e)) before its goal was attained in 1995 by Apostoli and Brown in Apostoli and Brown (1995b), and also independently, algebraically, by Urquhart in Urquhart (1995). Apostoli and 47

CHAPTER 4. NEIGHBORHOOD STRUCTURES

48

Brown’s proof was subsequently simplified, in 2000, by Nicholson, Jennings and Sarenac in Nicholson et al. (2000a). But both proofs exploit the compactness of colouring for hypergraphs whose edges are finitely long. Chromatic compactness, also called colouring compactness, is the claim that 8H , if H is a hypergraph then H is k-colourable .k  1/ iff every finite G  H is k-colourable.1 A hypergraph is a family of sets, called edges. A hypergraph H is k-colourable iff there is a partition of its union, called its vertex set, into k pairwise disjoint, mutually exhaustive sets, or cells, such that no edge of H is a subset of any cell. The simplifying thrust of Nicholson et al. in Nicholson et al. (2000a) raises the question whether there is an even simpler completeness proof, one which avoids citing chromatic compactness. In this paper an affirmative answer to this question is demonstrated, by invoking the theory of what are dubbed hyperframes. The theory of hyperframes implements the theory of transverse hypergraphs and consists essentially of a dualization of the neighborhood semantics for modal logic explored by Segerberg in Segerberg (1971), and referred to as ‘minimal models’ by Chellas in Chellas (1980).

4.2 Hyperframes A hyperframe F is pair .U; H/ where U is a non-empty set (the universe of the frame) and H is a hypergraph function from U to P P .U/. Accordingly, for each x 2 U; H.x/ is a hypergraph on U: a family of subsets of U, where the subsets are called the edges of the hypergraph, and the elements of the edges are called the vertices of the hypergraph. For each x in U, H.x/ is called the hypergraph on x (relative to F). A hyperframe is thus, in essence, a neighborhood frame, as the latter is defined in Segerberg (1971), for example. But a model on a hyperframe is distinct from a model on a neighborhood frame when it comes to interpreting . If F D .U; H/ is a hyperframe and V W Nat ! P .U/ is a valuation function, then .U; H; V/ is a (hyper)model M on F. Truth at a point x in a model M D .U; H; V/, with respect to the language of a standard 1 Colouring

compactness is provable from the compactness of propositional logic.

4.2. HYPERFRAMES

49

ˇ u

H.x/

x

kˇk

Figure 4.1: The truth condition for  on hyperframes.

propositional logic, is defined in the standard way for Boolean connectives. To interpret the unary necessity operator , we use the notion of the transversal of a hypergraph. Definition 1. If H is a hypergraph on a non-empty set U, and S is a subset of U, S is a transversal for H iff 8E 2 H; S \ E ¤ ;. It follows from this definition that ; 2 H iff H has no transversals, and H D ; iff, vacuously, every subset of U is a transversal for H. The truth condition for  is: ˆx ˇ iff kˇkM is a transversal for H.x/: M

(See Figure 1.) Introducing Þˇ as an abbreviation for ::ˇ, we therefore have: ˆx Þˇ iff 9E 2 H.x/ W kˇkM  E M

M If M D .U; H; V/ is a model and 8x 2 U; ˆ x ˛, then ˛ is valid on M, M written ‘ˆ ˛’; if for every valuation function V, ˛ is valid on .U; H; V/

CHAPTER 4. NEIGHBORHOOD STRUCTURES

50

then ˛ is valid on F D .U; H/, indicated by ‘F ˆ ˛’. If ˛ is valid on every member of a class C of frames, then ˛ is valid with respect to C, ‘C ˆ ˛’. The logic determined by the class C of all hyperframes is axiomatized by Nq ,2 the system defined by: ŒRR W ` ˛  ˇ ) ` ˛  ˇ

(4.1)

ŒPL W `PL ˛ ) ` ˛

(4.2)

ŒMP  W ` ˛  ˇ & ` ˛ ) ` ˇ ŒUS  W ` ˛ & ˇ is a substitution instance of ˛ ) ` ˇ

(4.3) (4.4)

It is easy to check that Nq is sound with respect to C; to prove completeness we use a Henkin construction which capitalizes on the theory of transverse hypergraphs. Definition 2. Let H be a hypergraph on a set U. A set E  U is a minimal transversal for H if E is a transversal for H and 8E0  E; E0 is not a transversal for H. The transverse hypergraph for H, T .H/, or just T H for convenience, is the set of all minimal transversals for H. Proposition 1. For any hypergraph H on a set U, H D ; iff T H D f;g, and ; 2 H iff T H D ;. Proof. Œ) Assume that H D ;. Then every subset of U is a transversal for H; but then if E ¤ ;, 9E0  E such that E0 is a transversal for H. So every minimal transversal for H is empty. And, ; 2 T H because ; has no proper subsets. Thus T H D f;g. Œ( Assume now that T H D f;g. Then 8E 2 H; ; \ E ¤ ;. Therefore H D ;. Œ) Suppose that ; 2 H. Then 8E 2 T H, E \ ; ¤ ;. Whence T H D ;. Œ( Lastly, suppose that T H D ;. If H D ; then by the above reasoning ; 2 T H. So H ¤ ;. If, then, ; 62 H, 9E¤; such that E 2 T H, which is absurd. Therefore ; 2 H.  Definition 3. A hypergraph H is simple if 8E; E0 2 H, E 6 E0 : 2 The

name of this system has been drawn from Jennings and Schotch (1981b).

4.2. HYPERFRAMES

51

Proposition 2. For any hypergraph H on a set U, T T H  H. If H is simple then H D T T H. Proof. Let H be a hypergraph on U, and let E 2 T T H. Suppose that E 62 H. Note that 8E0 2 H; E0 is a transversal for T H. Therefore 8E0 2 H; 9x 2 E0 such that x 62 E, in which case 9E0 2 T H such that E \ E0 D ;, contrary to the assumption that E 2 T T H. Therefore T T H  H. Now let E 2 H, and assume that H is simple. Since E is a transversal for T H, 9E0  E such that E0 2 T T H. But the above reasoning shows that T T H  H. Therefore, since H is simple, E0 D E, i.e., E 2 T T H, whence H  T T H.  If L is a modal logic and ˛ is a sentence, the proof set for ˛ in L, j˛jL , is the set of all maximal L-consistent sets of which ˛ is a member. The canonical frame for L is the structure FL D .UL ; HL / where UL is the class of all maximal L-consistent sets of formulae, and 8x 2 UL , HL .x/ D T .fj jL W  2 xg/: The canonical model for L, ML , is the triple .UL ; HL ; VL / where VL is defined: 8n 2 Nat; x 2 VL .n/ iff pn 2 x: Theorem 1. Let L be any logic closed under ŒRR; ŒPL; ŒUS and ŒMP . Then 8˛; x 2 UL ; k˛kML D j˛jL : Proof. We show that 8x 2 UL ; ˛ is true at x iff ˛ 2 x. The proof is by induction on the complexity of ˛. We omit all but the case for ˛ D ˇ. Suppose that x 2 kˇk. Then kˇk is a transversal for HL .x/, in which case, by the hypothesis of induction, so is jˇj. Therefore 9E  jˇj such that E 2 T .HL .x//. Whence E 2 fj j W  2 xg (Proposition 2). So let E D j j. Then j j  jˇj and  2 x. But then `  ˇ, and thus `   ˇ (by ŒRR). Therefore ˇ 2 x.

52

CHAPTER 4. NEIGHBORHOOD STRUCTURES

Suppose now that ˇ 2 x. Then 8E 2 HL .x/; E \ jˇj ¤ ;. I.e., jˇj is a transversal for HL .x/. By the induction hypothesis, jˇj D kˇk. Whence x 2 kˇk.  Corollary 1. The system Nq is determined by the class of all hyperframes.

4.3

Normal Hyperframes

The logic Nq is not normal because there is at least one theorem of Nq whose  formula is not a theorem. E.g., `Nq > while 6`Nq >. Since there is a hypermodel M containing a point x such that ; 2 H.x/, it follows that C 6ˆ >, and hence 6`Nq > (Corollary 1). By similar, dual, M reasoning there is a model M containing a point x such that 8˛; ˆ x Þ˛, M and thus ˆx Þ?. Hyperframes therefore provide an opportunity for the systematic investigation of non-normal logics, that is, logics that are not closed under the rule: ŒRN  W ` ˛ ) ` ˛

(4.5)

Definition 4. If F D .U; H/ is a hyperframe then F is normal if 8x 2 U; ; 62 H.x/. A model is normal if it is based on a normal hyperframe. This is significant from a philosophical perspective for two reasons: First, non-normal logics have deontic motivations insofar as we would like to not have an infinite number of obligations. A logic is normal when ˛ is a theorem whenever ˛ is a theorem. Thus, if  represents ‘it is obligatory that’, then in any deontic logic with an infinite number of theorems there is an infinite number of obligations.3 Second, if we read  as a necessity operator, then the existence of determined 3 It is important to note, however, that the absence of normality does not guarantee the absence of an infinite number of obligations. Although the absence of normality is necessary, it is not sufficient for this end. What we really need is the rule:

` ˛ ) ` :˛:

(4.6)

Jennings has suggested that what we really want is a variety of connexivist implication, which is a restriction of classical logic to contingencies.

4.3. NORMAL HYPERFRAMES

53

non-normal modal logics marks a conceptual divergence between logical validity and its classical, Aristotelian account. According to the classical account, an argument is valid when it is necessary that if the premises are true then the conclusion is true. But there are non-normal logics in which there are logically valid conditionals whose  formulae are not theorems. This raises the philosophical question of how theoremhood in such systems should be understood, or alternatively, the question of what the  operator represents. Theorem 2. Let L be a logic which is closed under ŒRR; ŒRN ; ŒUS ; ŒPL, and ŒMP . Then the canonical hyperframe for L is normal. Proof. Let x 2 UL , and suppose that ; 2 HL .x/. Then by Proposition 1, T HL .x/ D ;. But HL .x/ D T .fj j W  2 xg/; therefore T HL .x/ D fj j W  2 xg D ; (Proposition 2). I.e., 8 ;  62 x. But ` >, and so by ŒRN , ` >, in which case > 2 x, an absurdity.  .n C 1/-ary Relational Frames and n-Bounded Hyperframes An .n C 1/-ary relational frame .n  1/ is a pair .U; R/ where U is a non-empty set and R  UnC1 . If V is a function from Nat to P .U/ then the triple .U; R; V/ is an .n C 1/-ary relational model based on the frame .U; R/. Truth and validity relative to .n C 1/-ary relational frames and models are as defined for hyperframes and models, with the exception that truth at a point x for  formulae is defined: M

M

ˆx ˛ iff 8hy1 ; :::; yn i 2 R.x/; 9i 2 Œn Wˆyi ˛; where for any positive integer n, Œn denotes f1; 2; :::; ng, and if R  UnC1 , then R.x/ D fhy1 ; :::; yn i W hx; y1 ; :::; yn i 2 Rg. (See Figure 2.) Definition 5. A hyperframe F D .U; H/ is n-bounded, for n  1, if 8x 2 U; 8E 2 H.x/; jEj  n. A model is n-bounded if it is based on an n-bounded hyperframe.

CHAPTER 4. NEIGHBORHOOD STRUCTURES

54

˛

w1 w2

x z

n=3

zy1 ˛

z

z y2

z w3

z y3

z ˛ z3

z z1

z z z2 ˛

Figure 4.2: The truth condition for  on .n C 1/-ary relational frames.

Definition 6. Let F D .U; H/ be an n-bounded normal hyperframe, and let the relation R  UnC1 be defined pointwise: R.x/ WD fhy1 ; :::; ym ; yi ; :::; yi i W „ ƒ‚ … n m ti mes

fy1 ; :::; ym g 2 H.x/ & i 2 Œmg Then the .n C 1/-ary relational transformation of F is the .n C 1/-ary relational frame F D .U; R/. It is easy to see that for any model M on an n-bounded normal hyperframe there is an equivalent .n C 1/-ary relational model. That is: Theorem 3. For every normal n-bounded hyperframe F D .U; H/, every model M D .U; R; V/ on the .n C 1/-ary relational transfor-

4.4. WEAKLY AGGREGATIVE MODAL LOGIC

55

mation F D .U; R/ of F is pointwise equivalent to the hypermodel M D .U; H; V/, that is, M

M

8x 2 U; 8˛; ˆx ˛ ,ˆx ˛ Proof. The (omitted) proof is by induction on the complexity of ˛. 

4.4

Weakly Aggregative Modal Logic

A consequence of the theory of n-bounded normal hyperframes is a simple proof of the completeness of the system Kn with respect to the class of all .n C 1/-ary relational frames. The system Kn .n  1/ is defined: ŒRR W ` ˛  ˇ ) ` ˛  ˇ

(4.7)

ŒRN  W ` ˛ ) ` ˛

(4.8)

ŒPL W `PL ˛ ) ` ˛

(4.9)

ŒMP  W ` ˛  ˇ & ` ˛ ) ` ˇ

(4.10)

ŒUS W ` ˛ & ˇ is a substitution instance of ˛ ) ` ˇ _ ŒKn  W ` ˛1 ^ ::: ^ ˛nC1   ˛i ^ ˛j

(4.11) (4.12)

1i<j nC1

K1 is just the Kripke system K. For each n > 1, Kn is weakly aggregative because it replaces the strong aggregation principle ŒK.D ŒK1 / with the weaker ŒKn . It would appear that the completeness proof herein is a simplification of the other proofs in the literature, found in Apostoli and Brown (1995b) and Nicholson et al. (2000a), as both of these rely heavily on the colouring theory of hypergraphs, reference to which is omitted in the present proof. In particular, the previous proofs exploit colouring compactness, the claim that if H is a hypergraph each of whose edges is finitely long, then H is k-colourable iff every finite subgraph of H is k-colourable. In contrast, the crucial lemma used here, in addition to Theorem 2, is that the canonical hyperframe for any logic that includes ŒKn  .n  1/ is n-bounded.

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CHAPTER 4. NEIGHBORHOOD STRUCTURES

For convenience, we introduce the conventionSthat if t1 ; t2 ; :::; tnC1 2 are sets .n  1/, then nC1 .ti /i2ŒnC1 denotes 1i<j nC1 ti \ tj , 2 .˛i /i2ŒnC1 represents and if ˛1 ; ˛2 ; :::; ˛nC1 are sentences then nC1 W 1i<j nC1 ˛i ^ ˛j . Lemma 1. 8n  1, if H is a hypergraph such that 8E 2 H; jEj  n 2 .ti /i2ŒnC1 . then whenever t1 ; t2 ; :::; tnC1 are transversals for H, so is nC1 Proof. Suppose that 8E 2 H, jEj  n and that 8i 2 Œn C 1; ti is a 2 transversal for H. Suppose further that nC1 .ti /i2ŒnC1 \ E D ;, for some E 2 H. By a pigeonhole argument, 9i; j 2 Œn C 1.i ¤ j / such 2 .ti /i2ŒnC1 .  that ti \ tj \ E ¤ ;, which is absurd since ti \ tj  nC1 Lemma 2. Let H be a simple hypergraph such that 9n  1; 9E 2 H; jEj > n. Then there are n C 1 transversals for H, t1 ; t2 ; :::; tnC1 , 2 such that nC1 .ti /i2ŒnC1 is not a transversal for H. Proof. Let H be a simple hypergraph and let n  1 be an arbitrary integer such that for some E 2 H; jEj > n. Suppose that E D fx1 ; x2 ; :::; xi ; :::; xnC1 ; :::g. Then it is possible to construct n C 1 transversals for H, t1 ; t2 ; :::; tnC1 such that 8i 2 Œn C 1, ti \ E D fxi g; otherwise 9E0 2 H such that E0  E, contrary to the assumption that H 2 .ti /i2ŒnC1 \ E D ;. is simple. But then nC1  Theorem 4. 8n  1, if L is a modal logic which is closed under ŒRR, ŒPL, ŒUS , and modus ponens, then if ŒKn  2 L, it follows that the canonical frame for L is n-bounded. Proof. Assume that L is a modal logic which satisfies the antecedent conditions, including that ŒKn  2 L for some n  1. We show that 8x 2 UL , 8E 2 HL .x/, jEj  n. Suppose not. Let x 2 UL be such that E 2 HL .x/ and jEj > n. From Lemma 2 it follows that there are n C 1 transversals for HL .x/, t1 ; t2 ; :::; tnC1 , such that 2 nC1 .ti /i2ŒnC1 is not a transversal for HL .x/. But since ti is a transversal for HL .x/ .i 2 Œn C 1/, 9Ei  ti such that Ei 2 T .HL .x//, i.e., from Proposition 2, Ei 2 fj j W  2 xg. So let Ei D j i j .i 2 Œn C 1/,

4.4. WEAKLY AGGREGATIVE MODAL LOGIC

57

where  i 2 x. Since ŒKn  2 L, and L is closed under uniform sub2 stitution,  1 ^ ::: ^  nC1   nC1 . i /i2ŒnC1 2 x. Therefore M

2 2 . i /i2ŒnC1 2 x, and thus ˆx L  nC1 . i /i2ŒnC1 (Theorem 1), in  nC1 2 M which case k nC1 . i /i2ŒnC1 k L is a transversal for HL .x/. But:

k

2 2 . i /i2ŒnC1 kML D .k i kML /i2ŒnC1 nC1 nC1 2 D .j i j/i 2ŒnC1 nC1 2 D .Ei /i2ŒnC1 nC1

2 2 And nC1 .Ei /i 2ŒnC1  nC1 .ti /i2ŒnC1 . Therefore not a transversal for HL .x/, which is absurd.

2 nC1 .Ei /i2ŒnC1

is 

Since 8n  1, Kn is sound with respect to the class of all normal n-bounded hyperframes (see Lemma 1), given Theorems 2 and 4 we have: Corollary 2. 8n  1, the modal system Kn is determined by the class of all normal n-bounded hyperframes. In closing we have: Theorem 5. 8n  1; the system Kn is complete with respect to the class of all .n C 1/-ary relational frames. Proof. Assume that ˛ is valid with respect to the class of all .n C 1/ary relational frames. Then ˛ is valid on the .n C 1/-ary relational transformation of the canonical hyperframe FKn for Kn (Theorems 2, 4). Therefore, by Theorem 3, ˛ is valid on the canonical model MKn for Kn , whence `Kn ˛. 

Five

Polyadic Modal Logics and Their Monadic Fragments K AM S ING L EUNG

AND

R.E. J ENNINGS

Abstract It is well known that the smallest normal modal logic, called K, is determined by the class of binary relational frames. The collection of normal modal logics can be expanded by considering polyadic modal languages, each possessed of an n-ary modal connective and interpreted in (n+1)-ary relational frames. We survey a number of such logics and their classes of frames, and give metatheorems of soundness, completeness, and definability. Finally a series of monadic logics are obtained by diagonalizing the n-ary modal connectives. These monadic fragments support aggregation principles which are weaker than that of K, thus making them more suitable than K for such applications as deontic reasoning.

5.1

Introduction

The originating pulse of virtually all of the work presented in this volume lay in the inexpressibility, within the received Kripkean modal systems, of one or two deontically fundamental distinctions. The first was the distinction between deontic conflict, the common enough moral 59

60

CHAPTER 5. POLYADIC MODAL LOGICS

pickle in which, through incompetence or inattention, or circumstance, we find ourselves forced to neglect one moral duty to fulfill another. The topic has confronted moral systematizers for at least two hundred years. John Stuart Mill evidently supposed that if obligations are objectively grounded, any apparent such conflict can be no more than the starting point of a moral calculation that would ultimately dissolve it. Any likeminded philosopher might find little to complain of in the deontic reading of the modal  of some of the weaker Kripkean systems as It ought to be the case that, where a claim of deontic conflict ˛ ^ :˛ (a conjunction of contradictory obligations) is equivalent to a claim of a single obligatory contradiction ?. Most philosophers are convinced of the principle embodied in the Kantian slogan ”Ought implies can”, representable in the weak form ŒCon :? and some are no doubt happy to learn that as a matter of logic, that fundamental principle, of itself, rules out moral conflict, being equivalent to ŒD p ! ::p. If it is already a matter of logic, then it would seem that the question as to whether obligations are objectively grounded need not be answered. For some, no doubt, that resolution is a vindication of the Kripkean analysis of deontic necessity. For such people, it is an early discovery of deontic logic that the principle (call it “Kant”) that ought implies can is equivalent to the principle that there are no moral conflicts, and allows the latter to be elevated to the status of “The Deontic Law”. But for deontic logicians, there can be no such repose; nor for moral philosophers who accept the Kantian proscription of unfulfillable single obligations, but who recognize the presence of pairs or triples or larger families of obligations that force us to a kind of moral triage. For such theorists even the question as to what principles of deontic reasoning can be relied upon to constrain deontic conclusions remains itself a deontic matter. Logic does not relieve us of the responsibility to sort out such messes; the existence of such messes forces us to choose among logics. For such theorists, the distinctness of the two cases is itself a matter of logic. Accordingly, no system that does not preserve it is adequate. For the healthy skeptic, every principle of logic can be interrogated as to its suitableness for a particular application. For the deontic case, this applies to all of the principles that the Kripke systems force upon us.

5.1. INTRODUCTION

61

The principle ŒK p ^ q ! .p ^ q/ is certainly suspect, but so are the principles of normality and monotonicity. The former is embodied in the rule of necessitation ŒRN ` ˛ H) ` ˛, which makes every theorem an obligation. There would seem to be wanting in the moral tradition, that no obligation is unshirkable. Indeed, when we reflect that language is an evolved biological characteristic of humans, and moral language in particular is likely the product of selective social pressures, the exclusion of logically warranted sentences has some point. It is precisely the contingency of social arrangements that presses us to preserve the language by which we promote some and discourage others. Again, the principle ŒRM ` ˛ ! ˇ H) ` ˛ ! ˇ (the rule of monotonicity), which takes all logically provable consequences of obligatory acts as themselves obligations, can be ruled out on the related grounds that theorems are among the logically provable consequences of every obligatory act. Thus the adoption of a rule excluding theorems from obligation, would seem to force some restriction on [RM] or adopt as the underlying propositional system, one weaker that the classical PL. Now all of logic is, within varying limits, negotiable. Systems of logic are artifacts, and can be fashioned variously to serve varying goals, howsoever local. Nevertheless, there remains such a thing as pure research, and all of current deontic logical research must yet come under that rubric. So there is some point to exploring systems and classes of systems that have no other desirable feature than that they admit or deny particular distinctions, whatever other, independent distinctions they neglect. We might explore such a system for no purpose other than to satisfy our curiosity or to expand our understanding. In doing so, we might do well to proceed gradually from the familiar to less familiar, retaining, for the sake of illumination, principles which we expect ultimately to reject. We gain another benefit from this Fabian approach, for we learn about relationships and dependencies among principles that are masked in the more particular setting, but become evident in the more general. The simple expedient of generalizing the semantic analysis of the  to n-ary relational frames, substituting the binary truth-condition M; x ˆ ˛ ” .8y; Rxy H) M; y ˆ ˛/

CHAPTER 5. POLYADIC MODAL LOGICS

62

with the general M; x ˆ ˛ ” .8y1 ; : : : ; yn ; Rxy1


yn H) 9i.1  i  n/; M; yi ˆ ˛/ preserves both [RM] and [RN] as rules, but fails to validate [K]. Instead, it validate the considerably weaker ŒKn  p1 ^


^ pnC1 ! 

nC1 _

nC1 _

.pi ^ pj /:

iD1 j DiC1

In consequence, it distinguishes [D], which corresponds to no first-order restriction on n-ary R, and [Con], which corresponds to n-ary seriality: 8x; 9y1 ; : : : ; yn W Rxy1


yn : As we demonstrate below, the matter is not so simple as that. In the first place, although we do not discuss the matter here, a generalization to a set of n binary relations would serve to make at least some of the distinctions that a single binary relation conceals. And n-ary relational structures introduce their own conflations. Nor is the language of unary . the most natural or most expressive modal language for a study of n-ary relational structures. So we begin with a discussion of the correspondences between the language of an (n-1)-ary modal connective and the language of a single n-ary relation. Again, once we have seen the greater generality, it becomes evident that such a language generates more than one monadic fragment. In this introductory essay, however, we consider only the diagonal fragments of n-ary systems.

5.2

Polyadic modal languages and multi-ary relational frames

In this section, we introduce a series of modal languages, each of which extends the language of propositional logic with an n-ary modal operator. The frames and models for interpreting these languages are then described.

5.2. POLYADIC MODAL LANGUAGES AND MULTI-ARY RELATIONAL FRAMES

63

Definition 1. The n-adic modal language, denoted Ln , has the following symbols: denumerably many propositional variables (p1 , p2 , p3 , . . . ), connectives (?, :, _, and n ), and punctuations (the left and right parentheses ( and ), and the comma ,). The set of Ln -formulas is specified by the following rule in BNF: ˛ WWD pj?j:˛j.˛1 _ ˛2 /jn .˛1 ; : : : ; ˛n / where p is any propositional variable. Note that ? is a nullary connective while :, _, and n are unary, binary, and n-ary connectives, respectively. Other connectives such as >, ^, !, $, and Þn (the dual of n ) are introduced as shorthand: > abbreviates :?. .˛ ^ ˇ/ abbreviates :.:˛ _ :ˇ/. .˛ ! ˇ/ abbreviates .:˛ _ ˇ/. .˛ $ ˇ/ abbreviates ..˛ ! ˇ/ ^ .ˇ ! ˛//. Þn .˛1 ; : : : ; ˛n / abbreviates :n .:˛1 ; : : : ; :˛n /. We mention here some of the meta-logical conventions adopted in this paper. Outermost parentheses are omitted when writing formulas. _ and ^ bind more strongly than ! and $. Throughout this paper, we use lower case letters from the Greek alphabet ˛, ˇ, , . . . to denote formulas, and upper case letters €, , †, . . . , to denote sets of formulas. We write 8, 9, :, &, H) , and ” (or iff) for “for every”, “there exists”, “it is not the case that”, “and”, “(if . . . ) then”, and “if and only if”, respectively. Other meta-logical symbols will be introduced when needed. Definition 2. An n-ary relational frame F is a duple hU; Ri, where U , the universe of the frame, is a non-empty set of points, and R

CHAPTER 5. POLYADIC MODAL LOGICS

64

an n-ary relation on U . Given an n-ary relational frame F D hU; Ri, an n-ary relational model M is a duple hF; V i (or a triple hU; R; V i) where V is a valuation, i.e. a function assigning each propositional variable a set of points in U .

Definition 3. An Ln -formula ˛ is said to be true at a point x in an .n C 1/-ary relational model M D hU; R; V i (notation: M; x ˆ ˛) according to the following set of inductive conditions (where M; x 6ˆ ˛ means that ˛ is false at x in M): For every propositional variable pi , M; x ˆ pi iff x 2 V .pi /. M; x 6ˆ ?. M; x ˆ :˛ iff M; x 6ˆ ˛. M; x ˆ ˛ _ ˇ iff M; x ˆ ˛ or M; x ˆ ˇ. M; x ˆ n .˛1 ; : : : ; ˛n / iff 8y1 ; : : : ; yn ; Rxy1


yn H) 9yi .1  i  n/ W M; yi ˆ ˛i .

Given the above definition of Þn , we have the following condition: M; x ˆ Þn .˛1 ; : : : ; ˛n / iff 9y1 ; : : : ; yn W Rxy1


yn & 8yi .1  i  n/; M; yi ˆ ˛i . Truth conditions for the other defined connectives are straightforward and are omitted here. Definition 4. An Ln -formula ˛ is said to hold in a model M (notation: M ˆ ˛) if for every x in M, we have M; x ˆ ˛. Moreover ˛ is said to be valid on a frame F (notation: F ˆ ˛) if it holds in every model on F. If it is valid on every frame in a class C

5.3. NORMAL POLYADIC MODAL LOGICS

65

of frames, we say that it is valid on C (notation: C ˆ ˛). If ˛ is valid on the class of all frames, we simply say that ˛ is valid (notation: ˆ ˛). The presentation of formulas of an n-adic modal language and properties of an n-ary relational frame will be made simpler by adopting the following shorthand. Notation 1. 1. We often drop the subscripts of n and Þn since their arities are obvious from the number of propositions that are within their scope. 2. pE is an n-termed sequence p1 ; p2 ; : : : ; pn , and pEi is an n-termed sequence pi;1 ; pi;2 ; : : : ; pi;n . Similarly for xE and xE1 etc. 3. ?k is a k-termed sequence of ?’s, and similarly for >k .

5.3

Normal polyadic modal logics

A formal system S (in an object language) is essentially a proof-theoretic entity: it consists of a decidable set of formulas (called its axioms) and a finite set of inferential rules, each of which specifies what formula (conclusion) can be derived from other formulas (hypotheses). A formula ˛ is said to be provable from a set † of formulas (called assumptions) in S if there is a finite sequence of formulas, with the last member being ˛, and each member of the sequence being an axiom, or an assumption, or the conclusion of a rule of inference applied to some previous formulas in the sequence. We call such a sequence a proof of ˛ from † in S, and record its existence by writing † `S ˛ or simply † ` ˛ if the system is clear in the context. If ˛ is provable from the empty set of formulas, we call it a theorem of S and write `S ˛ or simply ` ˛ if the system is obvious. S is said to be inconsistent if the false is a theorem of it (`S ?), consistent if not (6`S ?). A logic, in comparison with a system, is usually characterized as a set of formulas (in an object language) that is closed under certain rules

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66

of inference. Given our definition of formal systems above, the set of theorems of a formal system constitutes a logic, but the converse is not generally true: there are logics which cannot be axiomatized as formal systems. Since we deal with axiomatizable logics only in this paper, we will no longer make the distinction between a system and a logic, and use the terms interchangeably from now on. Definition 5. An n-adic modal logic (in the modal language Ln ) is a consistent system that has the set of all the tautologies of propositional logic (or a suitable subset of it) included in its set of axioms, and the following rules in its set of inferential rules: ŒMP ŒUS

˛; ˛ ! ˇ ˛ `˛ ` ˛Œpi =ˇ

where ˛Œpi =ˇ is the formula that results from substituting ˇ for every occurrence of pi in ˛. Note that an n-adic modal logic (or simply an n-adic logic) is an extension of propositional logic in the sense that its set of theorems includes all the tautologies of propositional logic and is closed under the rules of modus ponens and uniform substitution. We express the same thing by saying that an n-adic logic includes PL. (Here PL refers to the set of tautologies together with the rules ŒMP and ŒUS.) Since the “early” days of modern modal logic, logicians have worked on monadic systems characterized as “normal”. Normality in this context means that the systems or logics have the following rules and axioms in addition to PL. `˛!ˇ ` ˛ ! ˇ `˛ ŒRN ` ˛ ŒK p ^ q !  .p ^ q/

ŒRM

5.3. NORMAL POLYADIC MODAL LOGICS

67

The smallest normal (monadic) system is called K (after Kripke). Various logics are obtained by adding axioms to K. KCon W ŒCon KD W ŒD KT W ŒT KB W ŒB K4 W Œ4 K5 W Œ5

:? p ! Þp p ! p p !Þp p ! p Þp !  Þ p

In what follows, we generalize normal monadic logics to normal nadic logics where n is a positive integer. Definition 6. An n-adic logic is called normal if it has in addition to PL the following schemas of rules of inference and axioms (where 1  i  n, and ˇ, q, and pi ^ q occur in the ith argument place of  as ˛i and pi do). ŒRMin  ŒRNin 

` ˛i ! ˇ ` .˛1 ; : : : ; ˛i ; : : : ; ˛n / ! .˛1 ; : : : ; ˇ; : : : ; ˛n / ` ˛i ` .˛1 ; : : : ; ˛i ; : : : ; ˛n /

ŒGin  .p1 ; : : : ; pi ; : : : ; pn / ^ .p1 ; : : : ; q; : : : ; pn / ! .p1 ; : : : ; pi ^ q; : : : ; pn /

We mention here an alternative way to characterize normal logics which makes use of PL, ŒRNin , and the following schema of axioms ŒGŠin . ŒGŠin  .p1 ; : : : ; pi ! q; : : : ; pn / ! ..p1 ; : : : ; pi ; : : : ; pn / ! .p1 ; : : : ; q; : : : ; pn // Note that there are n instances of each of ŒRMin , ŒRNin , ŒGin , and ŒGŠin . We shall refer to any one of them, or all of them, simply as ŒRMn , ŒRNn , ŒGn , and ŒGŠn , respectively.

CHAPTER 5. POLYADIC MODAL LOGICS

68

We call the smallest normal n-adic logic Gn . Note that G1 is just K. The name “Gn ” comes from D.K. Johnston, who names the system after Goldblatt for his introducing what amounts to G2 in an unpublished paper “Temporal Betweenness” (see ?). Other names have also been used in the literature: E Œn (E for entailment) in ?, K where  is a modal similarity type in Blackbutn et al. (2001). Various normal n-adic logics can be obtained by adding to Gn axioms which are generalizations of their monadic counterparts.

Definition 7. The following normal n-adic systems extend Gn with the indicated axioms. (In what follows, ) Gn Conn W ŒConn  :?n Gn Dn Gn Tn Gn Bn

W W W

ŒDn  pE ! ŒTn  pE ! ŒBn 

n ^

Gn 5n

W W

Þ .>i

Œ4n  pE ! Œ5n  ÞpE !

1

; p i ; >n i /

iD1 n _

pi

iD1 n ^

pi !

i D1

Gn 4n

n _

.?i

iD1 n ^

.?i

iD1 n ^

.?i

1

; Þp; E ?n i /

1

; p; E ?n i /

1

; Þp; E ?n i /

iD1

Each of the axioms and rules of inference mentioned previously has a dual form, which is logically equivalent to it (in any logic which provides propositional logic). We list the dual forms here for future refer-

5.3. NORMAL POLYADIC MODAL LOGICS

69

ence. ŒRMÞin  ŒRNÞin 

` ˛i ! ˇ ` Þ.˛1 ; : : : ; ˛i ; : : : ; ˛n / ! Þ.˛1 ; : : : ; ˇ; : : : ; ˛n / ` ˛i ` : Þ .˛1 ; : : : ; :˛i ; : : : ; ˛n /

ŒGÞin  Þ.p1 ; : : : ; pi _ q; : : : ; pn / ! Þ.p1 ; : : : ; pi ; : : : ; pn / _ Þ.p1 ; : : : ; q; : : : ; pn / ŒGÞin  : Þ .p1 ; : : : ; pi ; : : : ; pn / ! .Þ.p1 ; : : : ; q; : : : ; pn / ! Þ.p1 ; : : : ; :pi ^ q; : : : ; pn // ŒConÞn  Þ>n ŒDÞn  ŒTÞn  ŒBÞn  Œ4Þn  Œ5Þn 

n ^ iD1 n ^ iD1 n _ iD1 n _ iD1 n _

.?i

1

; pi ; ?n i / ! ÞpE

pi ! ÞpE Þ.>

i 1

; p; E >

n i

/!

n _

pi

iD1

Þ.>i

1

; Þp; E >n i / ! ÞpE

Þ.>i

1

; p; E >n i / ! pE

iD1

To conclude this section, we list some of the theorems of normal nadic logics below. (To highlight the axioms and rules used, we assume the base logic is just an n-adic logic.)

Proposition 1. The following are provable in any n-adic logic (i.e. logic in Ln that provides PL), given the specified rules and axioms.

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70

1. In the presence of ŒRMn  and ŒRNn : ŒBn  $

n ^ n _

pi;j ! .ÞpE1 ; : : : ; ÞpEn /




iD1 j D1

2. In the presence of ŒRMn : ŒBn  ! ŒGn  3. In the presence of ŒTn  and Œ4n : pE $ .?i

1

; p; E ?n i /

1

; Þp; E ?n i /

where 1  i  n. 4. In the presence of ŒTn  and Œ5n : pE $ .?i where 1  i  n. 5. In the presence of ŒRMn  and ŒGn : ŒConn  ! ŒDn  6. In the presence of ŒRNn : ŒDn  ! ŒConn  We leave the proof of the above proposition to the reader. Its import is as follows: (1) provides an alternative formulation of our ŒBn  in systems that have ŒRMn  and ŒRNn ; (2) is a generalization to the nary  of the result reported in ? for the unary ; (3) and (4) show that Œ4n  and Œ5n  are in some sense reduction axioms (in systems that have ŒTn ); (5) and (6) show that ŒConn  and ŒDn  are indeed equivalent in normal n-adic systems. Observe that what has been called ŒDn  here is weaker than the following formula. ŒDŠn  pE ! ÞpE

5.4. FRAME DEFINABILITY

71

We remark here that both ŒD1  and ŒDŠ1  are the same formula, viz. ŒD p ! Þp. So ŒDn  and ŒDŠn  represent two different ways to generalize ŒD. In normal monadic logics, ŒD and ŒCon are inter-derivable. Bearing this in mind, our ŒDn  appears to be a better way to generalize ŒD than ŒDŠn  does. (This is also reflected in the correspondence result in the next section: while ŒDn  and ŒConn  correspond to seriality, ŒDŠn  is not even first-order definable.)

5.4

Frame definability

Proposition 2. The following modal formulas correspond to the indicated first-order conditions: ŒConn 

W .8x/.9y/Rx E yE

(Seriality)

ŒDn 

W .8x/.9y/Rx E yE

(Seriality)

ŒTn 

W .8x/Rxx


x

(Reflexivity)

ŒBn 

W .8x/.8y/.Rx E yE ! .8yi 2 y/Ry E i x


x/ (Symmetry)

Œ4n 

W .8x/.8y/.Rx E yE ! .8yi 2 y/.8E E z /.Ryi zE ! RxEz // (Transitivity)

Œ5n 

W .8x/.8y/.Rx E yE ! .8yi 2 y/.8E E z /.RxEz ! Ryi zE// (Euclideanness)

Proof. For correspondence between any one of the modal formulas listed above (say ˛) and the respective first-order condition (say ), we prove that a frame F validates ˛ if and only if it validates . The if direction is trivial and is omitted here. For the only if direction, we show that if F does not validate , then there is a model on F in which ˛ fails at some point in the model. For ŒConn , assume that F D hU; Ri is not serial. Then there exists an x in U which is not related to any n-tuple of points in U . Consider an arbitrary model M on F. By the truth condition for , any -formula is true at x in M. So M; x 6ˆ :?n . For ŒDn , assume, as in the case of ŒConn , that F D hU; Ri is not serial, and so some x in U is not related to any n-tuple of points in U . Consider an arbitrary M D hF; V i. Trivially M; x ˆ p, E but

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M; x 6ˆ Þ.>i 1 ; pi ; >n i / for any i such that 1  i  n. So M is a counter-model of ŒDn . For ŒTn , assume that F D hU; Ri is not reflexive, i.e. there is an x in U such that Rxx


x does not hold. Consider a model M D hF; V i with V .pi / D U fxg for every i between 1 and n. It is clear that M; x ˆ p, E since for any zE such that RxEz , at least one of zE, say zj , is not x, and so M; zj ˆ pj . On the other hand, M; x 6ˆ p1 and similarly for p2 ; : : : ; pn . So M; x 6ˆ ŒTn . For ŒBn , assume F D hU; Ri is not symmetric, i.e. there exist an x 2 U and a yE 2 U n such that Rx yE and for some yi 2 y, E it is false that Ryi x


x. Consider a model M D hF; V i with V .p1 / D fxg, . . . , and V .pn / D fxg. Clearly, M; x ˆ p1 ^


^ pn . But M; yi 6ˆ Þp, E for if M; yi ˆ ÞpE then there exists zE such that Ryi zE and M; zj ˆ pj for all j from 1 to n, which implies that zj D x and so Ryi x


x, contrary to assumption. So M; x 6ˆ .?i 1 ; Þp; E ?n i /. It thus follows that M; x 6ˆ ŒBn . For Œ4n , assume F D hU; Ri is not transitive, i.e. there exist x 2 U , yE 2 U n , and zE 2 U n such that Rx y, E Ryi zE (for some yi 2 y), E and :RxEz . Consider a model M D hF; V i with V .p1 / D U fz1 g, . . . , and V .pn / D U fzn g. We argue that M; x ˆ pE as follows: for an arbitrary w E 2 U n such that Rx w E and for some wj 2 w, E we have wj 6D zj and so M; wj ˆ pj . However, M; x 6ˆ .?i 1 ; p; E ?n i / since M; yi 6ˆ pE (the reason being that Ryi zE, M; z1 6ˆ p1 , . . . , and M; zn 6ˆ pn ). In other words, M; x 6ˆ Œ4n . For Œ5n , assume F D hU; Ri is not euclidean, i.e. there exist x 2 U , yE 2 U n , and zE 2 U n such that Rx y, E RxEz , and :Ryi zE, for some yi 2 y. E Consider a model M D hF; V i with V .p1 / D fz1 g, . . . , and V .pn / D fzn g. Clearly M; x ˆ Þp. E However M; x 6ˆ i 1 n i .? ; Þp; E ? / or equivalently M; x ˆ Þ.>i 1 ; :p; E ?n i /, since M; yi ˆ :p, E for which we argue as follows: assume arbitrary n w E 2 U such that Ryi w, E then wj 6D zj (for some wj 2 w) E and so M; wj 6ˆ pj , i.e. M; wj ˆ :pj . In other words, M; x 6ˆ Œ5n . 

Theorem 2. The classes of .n C 1/-ary relational frames for the follow-

5.5. SOUNDNESS AND COMPLETENESS

73

ing normal n-adic logics are as indicated. Gn Conn Gn Dn Gn Tn Gn Bn Gn 4n Gn 5n

W Serial frames W Serial frames W Reflexive frames W Symmetric frames W Transitive frames W Euclidean frames

Proof. The theorem follows directly from Proposition 2.

5.5



Soundness and completeness

A consequence of Theorem 2 is the soundness of those logics with respect to their classes of frames. In this section, we demonstrate that they are also complete. Our strategy of proving the completeness of a normal n-adic logic L with respect to a class C of .n C 1/-ary relational frames is to show that every set of Ln -formulas consistent in L has a model on a frame in C. In fact, for any normal modal logic, there exists a model that satisfies any consistent set of formulas. (We call this model the canonical model of the logic, and the corresponding frame its canonical frame.) Given this result, all that remains to show the completeness of L with respect to the class C of frames is to show that the canonical frame of L belongs to C. In the following, we first define the canonical model of a normal nadic logic. Before showing that the canonical model is indeed a model for any consistent set of formulas, we prove an existence lemma and a truth lemma pertaining to such a logic and its canonical model. (The proof for what we call the existence lemma here is based on ?. ? has another proof. For a more recent version, readers can check Blackbutn et al. (2001).) Definition 8. [Canonical frames and models] Let L be a normal n-adic logic (in modal language Ln ). The L-canonical model, denoted ML , is a triple hUL ; RL ; VL i where:

CHAPTER 5. POLYADIC MODAL LOGICS

74

UL is the set of all L-maximal consistent set of Ln -formulas. For every x, y1 , . . . , and yn , RL xyi


yn iff the following condition holds: .˛1 ; : : : ; ˛n / 2 x H) 9˛i .1  i  n/ W ˛i 2 yi : For every x, x 2 VL .pi / iff pi 2 x. We call the duple hUL ; RL i the canonical frame of L. Lemma 1 (Existence Lemma for normal n-adic logics). Let ML D hUL ; RL ; VL i be the canonical model of a normal n-adic logic L. For any point x 2 UL and Ln -formulas ˛1 ; : : : ; ˛n , if :.˛1 ; : : : ; ˛n / 2 x, then there exist y1 ; : : : ; yn 2 UL such that :˛1 2 y1 , . . . , and :˛n 2 yn , and RL xy1


yn . Proof. Assume :.˛1 ; : : : ; ˛n / 2 x. We show, by induction, that there exist y1 ; : : : ; yn 2 UL such that each yi (1  i  n) satisfies both of the following requirements. (E1) :˛i 2 yi : (E2) For any formulas 1 ; : : : ; i 1 , ˇ, if : 1 2 y1 , . . . , : i 1 2 yi 1 , and . 1 ; : : : ; i 1 ; ˇ; ˛i C1 ; : : : ; ˛n / 2 x, then ˇ 2 yi . For the existence of y1 , we first show that y10 defined by letting y10 D f:˛1 g [ fˇj.ˇ; ˛2 ; : : : ; ˛n / 2 xg is L-consistent. Assume, for reductio, y10 is not L-consistent. Then, for some ˇ1 ; : : : ; ˇm 2 fˇj.ˇ; ˛2 ; : : : ; ˛n / 2 xg, the following hold. fˇ1 ; : : : ; ˇm ; :˛1 g `L ? `L ˇ1 ^


^ ˇm ! ˛1 `L .ˇ1 ^


^ ˇm ! ˛1 ; ˛2 ; : : : ; ˛n / .ŒRNn / `L .ˇ V 1 ^


^ ˇm ; ˛2 ; : : : ; ˛n / ! .˛1 ; ˛2 ; : : : ; ˛n / .ŒGn / `L jmD1 .ˇj ; ˛2 ; : : : ; ˛n / ! .˛1 ; ˛2 ; : : : ; ˛n / .ŒGn /

5.5. SOUNDNESS AND COMPLETENESS

75

Since .ˇj ; ˛2 ; : : : ; ˛n / 2 x for every j , and x is maximal L-consistent, we have .˛1 ; ˛2 ; : : : ; ˛n / 2 x. But this is impossible, for by assumption :.˛1 ; ˛2 ; : : : ; ˛n / 2 x. Thus, by reductio, y10 is L-consistent and so has a maximal L-consistent extension y1 (by Lindenbaum’s Lemma). It is straightforward to see that y1 satisfies both requirements (E1) and (E2) (for i D 1). To demonstrate the existence of the other members of the series, viz., y2 ; : : : ; yn , assume that we already have y1 ; : : : ; yk 2 UL which satisfy (E1) and (E2) in place (where k < n). As in the case of y1 , we define 0 an initial set ykC1 that can be shown to have a maximal L-consistent extension ykC1 satisfying both (E1) and (E2). So let 0 ykC1 D f:˛kC1 g [ fˇj9 1 ; : : : ; k W : 1 2 y1 ; : : : ; : k 2 yk &

. 1 ; : : : ; k ; ˇ; ˛kC2 ; : : : ; ˛n / 2 xg: 0 To show that ykC1 is L-consistent, we assume otherwise. Then, for 0 some ˇ1 ; : : : ; ˇm 2 ykC1 f:˛kC1 g, the following hold.

fˇ1 ; : : : ; ˇm ; :˛kC1 g `L ? `L ˇ1 ^


^ ˇm ! ˛kC1 For each ˇj (1  j  m), there exist : j:1 2 y1 ; : : : ; : j:k 2 yk such that . j:1 ; : : : ; j:k ; ˇj ; ˛kC2 ; : : : ; ˛n / 2 x: Then by ŒRMn  and ŒGn  we get .

m _

m _

j:1 ; : : : ;

j D1

j:k ;

j D1

m ^

ˇj ; ˛kC2 ; : : : ; ˛n / 2 x:

j D1

Since ˇ1 ^


^ ˇm ! ˛kC1 2 x, we also have the following by ŒRNn  .

m _

j:1 ; : : : ;

j D1

m _

m ^

j:k ;

j D1

ˇj ! ˛kC1 ; ˛kC2 ; : : : ; ˛n / 2 x:

j D1

Thus by ŒGn  we have .

m _

j D1

j:1 ; : : : ;

m _ j D1

j:k ; ˛kC1 ; ˛kC2 ; : : : ; ˛n / 2 x:

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CHAPTER 5. POLYADIC MODAL LOGICS

W Note that : jmD1 j:1 2 y1 , since : j:1 2 y1 for all j (1  j  m), W and y1 is maximal L-consistent. Similarly, : jmD2 j:2 2 y2 , . . . , and Wm Wm : j D1 j:k 2 yk . But j D1 j:k 2 yk , since yk complies with our 0 requirement (E2). Hence we derive a contradiction. By reductio ykC1 is L-consistent, and so has a maximal L-consistent extension ykC1 . It is straightforward to check that ykC1 satisfies requirements (E1) and (E2) (for i D k C 1). We have now demonstrated the existence of y1 ; : : : ; yn 2 UL all of which satisfy requirements (E1) and (E2). It remains to show that RL xy1


yn . Assume that for any ˇ1 ; : : : ; ˇn , .ˇ1 ; : : : ; ˇn / 2 x, ˇ1 … y1 , . . . , ˇn 1 … yn 1 . Then :ˇ1 2 y1 , . . . , :ˇn 1 2 yn 1 . Since yn satisfies (E2), we have ˇn 2 yn . Thus RL xy1


yn according to the definition of RL . This completes our proof of the Existence Lemma.  Lemma 2 (Truth lemma for n-adic logics). Let L be a normal n-adic logic. For any Ln -formula ML ; x ˆ ˛ ” ˛ 2 x: Proof. The proof is by induction on ˛. In the following we show the modal case only. Let ˛ be .˛1 ; : : : ; ˛n /, and show that for an arbitrary x 2 UL , ML ; x ˆ .˛1 ; : : : ; ˛n / ” .˛1 ; : : : ; ˛n / 2 x by assuming the inductive hypothesis that the theorem holds for ˛1 , . . . , and ˛n . For the direction H), assume that .˛1 ; : : : ; ˛n / … x, i.e. :.˛1 ; : : : ; ˛n / 2 x. Then, by the existence lemma, there exist y1 ; : : : ; yn 2 UL such that :˛1 2 y1 , . . . , and :˛n 2 yn , and RL xy1


yn . Then for each i such that 1  i  n, ˛i … yi and by the inductive hypothesis ML ; yi 6ˆ ˛i . Thus ML ; x 6ˆ .˛1 ; : : : ; ˛n /, as desired. For the direction (H, assume that .˛1 ; : : : ; ˛n / 2 x. To show that ML ; x ˆ .˛1 ; : : : ; ˛n /, we consider arbitrary y1 ; : : : ; yn 2 U such that RL xy1


yn . Then by the definition of RL , ˛i 2 yi for some i where 1  i  n. It follows from the inductive hypothesis that M; yi ˆ ˛i , whence we conclude that ML ; x ˆ .˛1 ; : : : ; ˛n /. 

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77

Corollary 1. Let L be a normal n-adic logic. Then any L-consistent set of formulas is satisfiable in the canonical model of L. Proof. Let † be an L-consistent set of formulas. By Lindenbaum Lemma, † can be extended to a maximal L-consistent set x of formulas. But every formula in † is true at x in ML , the canonical model of L, according to the truth lemma. In other words, † is satisfiable in ML .  Theorem 3. The following normal n-adic logics are complete with respect to the indicated classes of .n C 1/-ary relational frames: Gn Gn Conn Gn Dn Gn Tn Gn Bn Gn 4n Gn 5n

W All frames W Serial frames W Serial frames W Reflexive frames W Symmetric frames W Transitive frames W Euclidean frames

Proof. For the completeness of a logic L with respect to a class C of frames, it suffices to show that every L-consistent set of formulas is satisfiable in a model on a frame in class C. In the case of a normal modal logic L, we need only show that its canonical model belongs to C, for every L-consistent set of formulas is satisfiable in the canonical model according to the above corollary. (In the following, we use ML and RL for the canonical model and relation of the modal logic in context.) For Gn . It suffices to note that the the canonical model of Gn is an .n C 1/-ary relational frame. For Gn Conn . Let x in ML be arbitrary. Since ŒConn  is in x, we have by the fundamental theorem ML ; x ˆ Þ>n . Hence the canonical relation is serial. For Gn Dn . Let x in ML be arbitrary. Since both ŒDn  and >n are in x, we have Þ>n in x as well. Hence the canonical relation is serial. For Gn Tn . To show that the canonical relation is reflexive, we assume .˛1 ; : : : ; ˛n / 2 x, where x in ML , and formulas ˛1 , . . . , and ˛n are arbitrary. Since ŒTn  is in x, we have ˛1 _


_ ˛n in x. Then at

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CHAPTER 5. POLYADIC MODAL LOGICS

least one of ˛1 , . . . , ˛n is in x. Hence RL xx


x by the definition of the canonical relation. For Gn Bn . We assume RL xy1


yi


yn where x, y1 ; : : : ; yn in ML are arbitrary, and show that RL yi x


x. This will follow if we can show that if ˛1 … x, . . . , and ˛n … x then .˛1 ; : : : ; ˛n / … yi . So assume ˛1 … x, . . . , and ˛n … x. Then :˛1 2 x, . . . , and :˛n 2 x. Then :˛1 ^


^ :˛n 2 x. Since ŒBn  2 x, we have .?i 1 ; Þ.:˛1 ; : : : ; :˛n /; ?n i / 2 x. Given the assumption that RL xy1


yn , we conclude Þ.:˛1 ; : : : ; :˛n / 2 yi , i.e. :.˛1 ; : : : ; ˛n / 2 yi . But this just means that .˛1 ; : : : ; ˛n / … yi , which is what we want. For Gn 4n . We assume RL xy1


yi


yn and RL yi z1


zn where x, y1 ; : : : ; yn , z1 ; : : : ; zn are arbitrary points in ML . To show that RL xz1


zn , we argue that for any formulas ˛1 ; : : : ; ˛n , if .˛1 ; : : : ; ˛n / 2 x, then ˛j 2 zj , for some j where 1  j  n. So assume .˛1 ; : : : ; ˛n / 2 x. Since Œ4n  2 x, we have .?i 1 ; .˛1 ; : : : ; ˛n /; ?n i / 2 x. Then .˛1 ; : : : ; ˛n / 2 yi , given that RL xy1


yn . Then, as desired, we have ˛j 2 zj for some j such that 1  j  n, given that RL yi z1


zn . For Gn 5n . We assume that RL xy1


yi


yn and RL xz1


zn where x, y1 ; : : : ; yn , z1 ; : : : ; zn are arbitrary. To show that RL yi z1


zn , we argue that for any formulas ˛1 ; : : : ; ˛n , if ˛j … zj , where j ranges from 1 to n, then .˛1 ; : : : ; ˛n / … yi . So assume for all j from 1 to n, ˛j … zj . Since RL xz1


zn , we have .˛1 ; : : : ; ˛n / … x, i.e. :.˛1 ; : : : ; ˛n / 2 x, i.e. Þ.:˛1 ; : : : ; :˛n / 2 x. Then .?i 1 ; Þ.:˛1 ; : : : ; :˛n /; ?n i / 2 x, given that Œ5n  2 x. Then Þ.:˛1 ; : : : ; :˛n / 2 yi , given that RL xy1


yn . In other words, .˛1 ; : : : ; ˛n / … yi , as desired. 

5.6

Diagonal fragments of normal polyadic modal logics

We introduce the unary operator  and its dual Þ into the n-adic modal language Ln by the following identities: ˛ D n .˛; : : : ; ˛/

Þ˛ D Þn .˛; : : : ; ˛/

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79

These unary operators can be described as “diagonalization” of their n-ary counterparts: consider the set of n-tuples of formulas as an ndimensional matrix, and the set of tuples whose coordinates are the same formula as a diagonal across such a matrix. The truth conditions for the diagonal operators are as follows: M; x ˆ ˛ iff 8y1 ; : : : ; yn ; Rxy1


yn H) 9yi .1  i  n/ W M; yi ˆ ˛. M; x ˆ Þ˛ iff 9y1 ; : : : ; yn W Rxy1


yn & 8yi .1  i  n/; M; yi ˆ ˛. Let Ln   be the set of Ln -formulas that can be expressed by using only the unary  and truth-functional connectives. We call the set of theorems of Gn that can be so expressed, i.e. the intersection of Gn and Ln  , the diagonal fragment of Gn . This fragment has been shown to be axiomatized by the following system Kn . For details, see ?, ?, and ?. Definition 9. Kn has PL, ŒRM, ŒRN, and the following axiom. ŒKn  p1 ^


^ pnC1 ! 

nC1 _

nC1 _

.pi ^ pj /

i D1 j DiC1

Note that K1 is just K. Kn can be described as a “weakly aggregative modal logic” since its aggregate principle ŒKn  is a weakening of the following principle of complete aggregation, which is a theorem of K. ˛1 ^


^ ˛n ! .˛1 ^


^ ˛n / Kn can be extended by adding the familiar formulas ŒK, ŒCon, ŒT, ŒB, Œ4, and Œ5, and the resulting systems are called, respectively, Kn K, Kn Con, Kn T, Kn B, Kn 4, and Kn 5. Note that Kn K is just K1 or K, since ŒKn  is derivable from ŒK. In the following, we list correspondence and determination results for these formulas and logics.

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80

Theorem 4. The following modal formulas correspond to the indicated first-order conditions. ŒK

W .8x/.8y/.Rx E yE ! .9yi 2 y/Rxy E i


yi / (Quasi-binarity)

ŒCon

W .8x/.9y/Rx E yE

(Seriality)

ŒT

W .8x/Rxx


x

(Reflexivity)

ŒB

W .8x/.8y/.Rx E yE ! .9yi 2 y/Ry E i x


x/

(SymmetryŽ)

Œ4

W .8x/.8y; E zE1 ; : : : ; zEn /.Rx yE ^ Ry1 zE1 ^


^ Ryn zEn ! .9w E  zE1 [


[ zEn /Rx w/ E

Œ5

(TransitivityŽ)

W .8x/.8y; E zE/.Rx yE ^ RxEz ! .9yi 2 y/Ry E E/ iz (EuclideannessŽ)

Note: In the condition of transitivity, 9w E  zE1 [


[ zEn means the following: there exists a w E such that every wk 2 w E belongs to the set of zi;j ’s where 1  i; j  n. Observe that the frame properties corresponding to ŒB, Œ4, and Œ5 are weaker than those corresponding to ŒBn , Œ4n , and Œ5n , which is expected since the former formulas are derivable from the latter ones. We distinguish the weaker properties from the stronger ones by the sign Ž. Theorem 5. The following diagonal logics are determined by the indicated classes of .n C 1/-ary relational frames. Kn W All frames Kn K W Quasi-binary frames Kn Con W Serial frames Kn T W Reflexive frames Kn B W SymmetricŽ frames Kn 4 W TransitiveŽ frames Kn 5 W EuclideanŽ frames

Six

Preserving What? G ILLMAN PAYETTE AND P ETER S CHOTCH

Abstract In this essay Gillman Payette and Peter Schotch present an account of the key notions of level and forcing in much greater generality than has been managed in any of the early publications. In terms of this level of generality the hoary notion that correct inference is truth-preserving is carefully examined and found wanting. The authors suggest that consistency preservation is a far more natural approach, and one that can, furthermore, characterize an inference relation. But an examination of the usual account of consistency reveals problems that, in general, can be corrected by means of an auxiliary notion of inference (forcing) which relies upon a kind of generalization of consistency, called level. Preservation of the latter is shown to be another of the properties which characterize a logic and forcing is shown to preserve it. The essay ends with a sketch of a result which locates forcing among all possible level-preserving inference relations.

6.1

Introduction

The (classical) semantic paradigm for correct inference is often given the name ‘truth-preservation.’ This is typically spelled out to the awestruck students in some such way as: 81

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82

An inference from a set of premises, €, to a conclusion, ˛, is correct, say valid, if and only if whenever all the members of € are true, then so is ˛. This understanding of the slogan may be tried, but is it actually true? There is a problem: the way that ‘truth’ is used in connection with the premises is distinct from the way that it is used with the conclusion. In other words, this could be no better than a quick and dirty gloss. The chief virtue of the formulation is that of seeming correct to the naive and untutored. But what of the sophisticates? They might well ask for the precise sense in which truth is supposed to be preserved in this way of unpacking. On the right hand side of the ‘whenever’ we are talking about the truth of a single formula while on the left hand side we are talking about the truth of a bunch of single formulas. Is it the truth of the whole gang which is ‘preserved?’ Of course it is open to the dyed-in-the-wool classicalist to reply scornfully that we need only replace the set on the left with the conjunction of its members. In this way is truth preserved from single formula to single formula as homogenously as anyone could wish. It is open, but not particularly inviting. In the first place, this strategy forces us to restrict the underlying language to one with conjunction and conditional connectives—that must operate in something like the usual (which is to say classical) way. There are enough who would chafe under this restriction that a sensitive theorist would hesitate to impose it. Apart from the objection, we are inclined to think of this business of coding up the valid inferences in terms of their ‘corresponding conditionals’1 as an accident of the classical way of thinking, and that it is no part of the definition of a correct account of inference. We also notice that on the proposal, we are restricted to finite sets. Setting aside this unpalatable proposal then, we ask how is this notion of truth-preservation supposed to work? Since there is no gang on 1 We

think this usage was coined first by Quine.

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83

the right we seem to be talking about a different kind of truth, individual truth maybe, from the kind we are talking about on the left—mass truth perhaps. Looked at in that somewhat jaundiced way, there isn’t any preserving going on at all, but rather a sort of transmuting. The classical paradigm really ought to be given by the slogan ‘truth transmutation.’ In passing from the gaggle of premises to the conclusion, gaggle-truth is transmuted into single formula truth. It may be more correct to say that, but it makes the whole paradigm somewhat less forceful or even less appealing. What we need in order to rescue the very idea of preservationism is to talk entirely about sets. So we shall have to replace the arbitrary conclusion ˛ with the entire set of conclusions which might correctly be drawn from €. We even have an attractive name for that set—the theory generated by € or the deductive closure of €. In formal terms this is C` .€/ D f˛j€ ` ˛g Now that we have sets, can we say what it is that gets preserved— can we characterize classical inference, for instance, as that relation between sets of formulas and their closures such that the property ˆ is preserved? We can see that gaggle-truth would seem to work here in the sense that whenever € is gaggle-true so must be C` .€/, for ` the classical notion of inference at least. We are unable to rid ourselves, however, of the notion that gaggle-truth is somewhat lacking from an intuitive perspective. Put simply, our notion of truth is carried by a predicate which applies to sentences, or formulas if we are in that mood. These are objects which might indeed belong to sets but they aren’t themselves sets. So however we construe the idea of a true set of sentences (or formulas) that construal will involve a stipulation or more charitably, a new definition. Generations of logic students may have been browbeaten into accepting: ‘A set of sentences is true if and only if each member of the set is true.’ But, it is a stipulation, and is no part of the definition of ‘true.’ It doesn’t take very much imagination to think that somebody might actually balk at the stipulation. Somebody who is attracted to the

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idea of coherence for instance, might well want to say that truth must be defined for (certain kinds of) sets first and that the sentential notion is derived from the set notion and not conversely. All of which is simply to say that a stipulation as to how we should understand the phrase ‘true set of sentences’ is unlikely to be beyond the bounds of controversy.2 It may gladden our hearts to hear then, that there is another property, perhaps a more natural one, which will do what we want. That alternative property is consistency.

6.2

Making A Few Things Precise

By a logic X , over a language L we understand the set of pairs h€; ˛i such that € is a set of formulas from the language L and ˛ is a formula from that same language, and € `X ˛. In the sequel we frequently avoid mention of the language which underlies a given logic, when no confusion will thereby be engendered. This set of pairs is also referred to as the provability or inference relation of X . In saying this we expose our extensional viewpoint according to which there is nothing to a logic over and above its inference relation. This has the immediate consequence that we shall take two logics X and Y which have the same inference relation, to be the same logic. When X is a logic, we refer to the X-deductive closure of the set € by means of ‘CX .€/’ Unless the contrary is specified, Every logic mentioned below will be compact, which is say that whenever € ` ˛ it follows that there must be some finite subset, say , of €, which proves ˛. In mentioning consistency, we have in mind some previously given notion of inference, say `X . Each inference relation spawns a notion of consistency according to the formula 2 It may be helpful here to consider an analogy between sentences and numbers, taken to be urelementen. We can define the idea of a prime number easily enough but be puzzled about how to define a prime set of numbers. Somebody might be moved to offer: ‘Why not simply define a prime set of numbers to be a set of prime numbers?’ The answer is likely to be: ‘Why bother?’ Indeed the whole idea of a prime set of numbers seems bizarre and unhelpful. We can easily imagine circumstances in which we would require a set of prime numbers but the reverse is true when it comes to a prime set.

6.2. MAKING A FEW THINGS PRECISE

85

€ is consistent, in or relative to a logic X (alternatively, € is X-consistent) if and only if there is at least one formula ˛ such that € °X ˛. To say this in terms of provability rather than non-provability we might issue the definition: € is inconsistent in a logic X if and only if CX .€/ D S, where S is the set of all formulas of the underlying language of X . Where X is a logic, the associated consistency predicate (of sets of formulas) for X, is indicated by CONX . We were interested in how an inference relation might be characterized in terms of preserving some property of sets. We have singled out consistency as a natural property of sets and having done that we can see that preservation of consistency comes very naturally indeed. The time has come to say a little more exactly what we mean by ‘characterized.’ In order to do this we shall be making reference to the following three structural rules of inference. [R] ˛ 2 € H) € ` ˛ [Cut] €; ˛ ` ˇ & € ` ˛ H) € ` ˇ [Mon] € ` ˛ H) € [  ` ˛ Unless there is a specific disavowal, every inference relation we consider will be assumed to admit these three rules. It should be noted that on account of [Mon], if the empty set ¿ is inconsistent in X, then the inference relation for that logic contains every pair h€; ˛i. In such a case we say that X is the trivial logic over its underlying language. We shall take the logics we mention from now on to be non-trivial, barring a disclaimer to the contrary. Let us say that an inference relation, say ’`X ’, preserves consistency if and only if:

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If € is X-consistent (in the sense of the previous definition), then so is CX .€/. It is easy to see that every inference relation with [Cut] and [Mon] must preserve consistency since if the closure of a set € proves some formula, ˛, then by compactness some finite sequence of [Cut] operations will lead to the conclusion that € proves ˛. It may be that we end up showing that some subset of € proves ˛, which is why we require [Mon] in this case. We say that X preserves consistency in the strong sense when the condition given above as necessary is also sufficient. It is similarly easy to see that since every set is contained in its deductive closure by [R], and since inconsistency is preserved by supersets, given [Mon], every inference relation satisfying the three structural rules preserves consistency in the strong sense. This is all very well, but we haven’t really gotten to anything that would single out an inference relation from among a throng of such, all of which preserve consistency. In order to do that it will be necessary to talk about a logic X preserving the consistency predicate of a logic Y , in the strong sense. A moment’s thought will show us that when the preservation is mutual, when X and Y preserve each other’s consistency predicates, (which implies that they share a common underlying language) then they must agree on which sets are consistent and which are inconsistent. For consider, if CONX .€/ and Y preserves the X consistency predicate then CONX .CY .€//. Suppose that € is not Y -consistent, then CY .€/ D S. By [R] CX .CY .€// D CX .S/ D S which is to say that CY .€/ is not X -consistent, a contradiction. Similarly for the argument that € is consistent in Y and X preserves the Y consistency predicate. When two logics agree in this way, i.e. agree on the consistent and inconsistent sets, we shall say that they are at evens. Another moment’s thought reveals that two logics which are at evens will preserve each other’s consistency predicates. Assume X and Y are at evens and CON X .€/ but CON X .CY .€//, where the overline indicates predicate negation. Then CONY .CY .€// because X and Y agree on inconsistent

6.2. MAKING A FEW THINGS PRECISE

87

sets. Hence by idempotentcy of CY € is not consistent in Y , a contradiction. Thus we have: Proposition 1. Any two logics X and Y over a language L are at evens if and only if X and Y preserve each other’s consistency predicates. This is nearly enough to guarantee that X and Y are the same logic. All we need is a kind of generalized negation principle: Definition 1. A logic X is said to have denial provided that for every formula ˛, there is some formula ˇ such that CONX .f˛; ˇg/. In such a case we shall say that ˛ and ˇ deny each other (in X , which qualification we normally omit when it is clear from the context). We will assume that any logic we mention has denial. Clearly if a logic has classical-like negation rules then it has denial, since the negation of a formula will always be inconsistent with the formula. Of course classically, there are countably many other formulas which are inconsistent with any given formula—namely all those which are self-inconsistent. The generalized notion doesn’t require that there be distinct3 denials for each formula, only that there be some or other formula which is not consistent with the given formula. Evidently, if two logics are at evens, then if one has denial, so does the other. In fact something stronger holds, namely: Proposition 2. If two logics X and Y are at evens, and X has denial then, for every formula ˛ there is some formula ˇ for which both CON X .f˛; ˇg/ and CON Y .f˛; ˇg/. Finally, we shall require of our logics that they satisfy the principle that denial commutes with provability in the right way: [Den] € `X ˛ ”

CON X .€

[ fˇg/ where ˇ denies ˛.

Now we are ready to state our result: 3 Distinct

up to logical equivalence, it goes without saying.

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CHAPTER 6. PRESERVING WHAT?

Theorem 1 (Generalized Consistency Theorem). Two logics X and Y are at evens if and only if, X and Y are the same logic. Proof. For this argument we split the equivalence into its necessary and sufficient halves. ( H) / Assume CX .€/ D CY .€/ for every set €—which is to say that X = Y . To say that CONX .€/ is to say that the X-closure of € is S. But then so must be the Y -closure of €. Similarly, to say that CONX .€/ is to say that there is some ˛ which is not in the X -closure of €, but then neither can ˛ be in the Y -closure of €, hence CONY .€/. So X and Y are at evens. ((H) Suppose then that X and Y are at evens. Let € be a consistent set, which means by the assumption, that it is consistent in both logics. Assume for reductio that € `X ˛ and € °Y ˛, and let ˇ deny ˛. Thus,by [Den] CONX .€ [ fˇg/ and CONY .€ [ fˇg/, a contradiction. 

6.3

What’s Wrong With This Picture?

To answer the question in the section heading, there really isn’t anything wrong with an approach which characterizes inference in terms of preserving consistency. It’s consistency itself, or at least many accounts of it, which casts a shadow over our everyday logical doings. The way we have set things up, a set € of formulas is either consistent in a logic X, or it isn’t. But it doesn’t take much thought to see that such an all-or-nothing approach tramples some intuitive distinctions. In particular, we may find the reason for the inconsistency to be of interest. In the logic X, for example, there may be a single formula ı which is, so to speak, inconsistent by itself. In other words CONX .fıg/. Formulas of this dire sort are sometimes described as being self-inconsistent (in X) or absurd in X. By [Mon] any set of formulas which contains a self-inconsistent formula is bound to be inconsistent.

6.4. SPEAK OF THE LEVEL

89

Thinking of the possible existence of absurd formulas leads us to sharpen our previous notion of denial: Definition 2. We shall say that X has non-trivial denial if and only if for every non-absurd formula ˛ there is at least one non-absurd formula ˇ which is not X -consistent with ˛. We are now struck by the contrast between X -inconsistent sets which contain X -absurdities and those which do not. Isn’t there an important distinction between these two cases? If we think of consistency as a desirable property which we are willing to trouble ourselves to achieve, then the trouble will be light indeed if all we need do is reject absurdities. On the other hand, an entire lifetime of angst may await those who wish to render consistent their beliefs or their obligations.4 There is a great deal more that one could say on this topic and some of the current authors have said much of it. For now, we shall take it that the need for a distinction has been established and our job is to construct an account of consistency which allows it. We have in mind building upon what we have already discovered instead of pursuing a slash-and-burn policy. This means, among other things, that the predecessor account should appear as a special (or limiting) case of the new proposal. The intuitive distinction bruited above, is clearly a distinction between different kinds of inconsistency, or perhaps different degrees. We might think of one kind being worse than the other, which leads to a rather natural way of classifying inconsistency.

6.4

Speak of the Level

The account of inconsistency which we propose is a generalization of the one first suggested in the 20th Century, in the work of Jennings and Schotch5 , namely the idea of a level (of incoherence, or inconsistency). 4 In

saying this, we assume that nobody is obliged to bring about anything impossible and that whatever is self-inconsistent cannot truly be a belief. 5 See especially Schotch and Jennings (1980c, 1989).

CHAPTER 6. PRESERVING WHAT?

90

The basic idea is that we can provide an intuitive measure of how inconsistent a set is by seeing how finely it must be divided before all of the divisions are consistent. What in the earlier account is stated in terms of classical provability, we now state in terms of arbitrary inference relations that satisfy the minimal conditions given in the earlier section. It all begins with the notion of a certain kind of indexed collection of sets being a logical cover for a set € of formulas, in the logic X . First we need a special kind of indexed family of sets (of formulas). ˚ Definition 3. A./ D a0 ; a1 ; : : : ; a is an indexed set starting with , provided a0 D  and all the indices 0 : : :  are drawn from some index set I .

Definition 4. Let F be an indexed set starting with ¿. F is said to be a logical cover of the set †, relative to the logic X , indicated by COV X .F; †/, provided: For every element a of the indexed family, CON X .a/ and S † CX .ai / i2I

So an X-logical cover for € is an indexed family of sets starting with the empty set, such that there are enough logical resources in the cover to prove, in the logic X, each member of €. Evidently, given the rule [R], f¿; €g will always be a logical cover of € if the latter set is X -consistent though it won’t, in general, be the least. If F† is a logical cover for the set †, the cardinality jI j 1 where I is the index set for F† , is referred to as the width of the cover, indicated by w.F† /. In the special circumstance that all the members of a logical cover of † are disjoint, the cover is said to partition †.6 6 This

should be contrasted with a covering family being a partition. We can recover the latter

6.4. SPEAK OF THE LEVEL

91

And finally we introduce the notion at which we have hinted since the start of this section. Definition 5. The level (relative to the logic X) of the set € of formulas of the underlying language of X, indicated by `X .€/ is defined: 8 < min ŒCOVX .F; €/ if this limit exists `X .€/ D w.F/ :1 otherwise

In other words: the X-level (of incoherence or inconsistency) of a set † in a logic X is the width of the narrowest X-logical cover of †, if there is such a thing, and if there isn’t, the level is set to the symbol 1. One might think that there will fail to be a narrowest logical cover when there is more than one—when several are tied with the least width, but this is a misreading of the definition. There might indeed be several distinct logical covers but there can only be one least width (which they all share). The uniqueness referred to in the definition attaches to the width, not to the cover, so to speak. The only circumstance in which there might fail to be a narrowest logical cover, is one in which † has no logical covers at all. In this circumstance † must contain what we earlier called an absurd formula. This notion satisfies both the requirement that it distinguishes between inconsistent sets which contain absurd formulas, and those which don’t, and the requirement that the predecessor notion of consistency relative to X appears as a special case. For it is clear that if € is an X -consistent set of formulas which does not consist entirely of X theorems, then a narrowest logical cover of € is f¿; a1 g where €  CX .a1 / and CONX .a1 /. So at least part of the earlier notion of CONX .€/ is captured by `X .€/ D 1. notion from this one by intersecting the covered set with each of the disjoint sets in the logical cover.

CHAPTER 6. PRESERVING WHAT?

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There is even an interesting insight which comes out of this new idea. For there are two levels of X-consistency, 0 and 1. In our earlier naive approach, we thought of consistency as an entirely monolithic affair, but once we give the matter some thought we see that the empty set does indeed occupy a unique position in the panoply of X -consistent sets. If we know only that € and † are both X -consistent, nothing at all follows about the X-consistency of € [†. But we may rest assured that both † [ ¿ and € [ ¿ are X -consistent. And the same goes for the Xconsequences of the empty set, namely X-theorems. By our definition, `X ./ D 0 if  is empty or any set of X-theorems, and of course such sets are consistent with any X-consistent set of formulas. It is tempting to call these level 0 sets hyperconsistent.

6.5

Level Preservation

So now that we have the concept of an X-level, should we be concerned about preserving such a thing? Perhaps there is no need for such concern, since it is at least possible that the logic X preserves its own level, isn’t it? Well, in a word, no. It is not in general true that X preserves level beyond, of course the levels 0 and 1 of X-consistency. All the logics we consider not only do that, but are characterized by doing that. Suppose the set € contains not only the formula ˛ but also a denial ˇ of ˛, although it does not contain any X -absurdities. Now, since X has the rule [R], € must X-prove both ˛ and ˇ. If X permits the arbitrary conjunction of conclusions, then € will prove an X -absurd formula, namely ˛ ^ ˇ. Hence by the meaning of closure, that absurdity belongs to the X-closure of €, which must thus have X -level 1. This amounts to a massive failure to preserve level. The obvious question to raise is this: given that `X is a generalization of CONX , is it the case that level characterizes logics in the same way that preserving consistency (in the strong sense) does? If not, then it seems that the generalization is not perhaps as central a notion as the root idea upon which it generalizes. Fortunately for supporters of the general notion, we may prove the following generalization of the Generalized Consistency Theorem.

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93

Theorem 2 (Level Characterization Theorem). Suppose that X and Y are inference relations over the same language L and let `X and `Y be the level functions associated with the respective inference relations. Then Œ`X .€/ D `Y .€/ for every set € of formulas of L ” X D Y . Proof. The proof depends upon the Generalized Consistency Theorem. ( H) ) Assume that `X .€/ D `Y .€/ for every set € of formulas of L. Then by definition the two agree on which sets have level 1 and level 0. But this is to say that X and Y agree on which sets are consistent. But by the Generalized Consistency Theorem, any two such logics (logics which we say are at evens) must be identical. ((H) Assume that X D Y and suppose that for some arbitrary set € of formulas of L `X .€/ D , for  some cardinal. Then, by the definition, there is a narrowest X -logical cover F€ such that w.F€ / D . Since X D Y , it must be the case by the Generalized Consistency Theorem that the two logics agree on consistency (in the strong sense). Further, by definition each ai 2 F€ is such that CONX .ai /. But then, since X and Y are at evens,CONY .ai /. Thus, F€ must be a Y -logical cover of € of width . Moreover, this must be the narrowest such logical cover or else by parity of reasoning, there would be an X -logical cover of cardinality less than  contrary to hypothesis. Since € was arbitrary it follows that `X and `Y must agree on all sets of formulas of the language L.  This suggests that level is worth preserving, that it is a sort of natural logical kind, but doesn’t show how the preservation may be carried out. It is time to repair that lack. Perhaps the most straightforward route to preserving X-level is to define a new inference relation based on X. Evidently the definition in question must also connect somehow with the notion of X-level and thus ultimately to CONX (from now on we shall mostly drop reference

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to the background logic, like X, when no confusion will result). The process might have been informed by the ancient joke: Question: How do you get down from an elephant? Answer: You don’t get down from an elephant, you get down from a duck. except in our case the question and answer would go: Question: How do you reason from inconsistent sets? Answer: You don’t reason from inconsistent sets, since every formula follows in that case, you reason from consistent subsets. In other words, an inconsistent set is one for which the distinction between what follows and what doesn’t has collapsed. This lack of meaningful contrast means that it no longer makes sense to talk about inferring conclusions from such a set. In order to regain the distinction we are going to have to drop back to the level of consistency and the only way to do that, is to look at consistent subsets of the original set. Absent the notion of level, there are different ways to do this. The one suggested in Quine and Ullian (1970) to deal with inconsistent sets of beliefs, involves two stages: At the first stage we discover the smallest subset of the inconsistent set which still exhibits the inconsistency. At the second stage we discard the member of the inconsistent subset with the least evidence, and repeat as necessary until the set is consistent. Having thus cleansed the belief set, we may now draw conclusions as we did before.7 We don’t say that this process can’t work. We do say that it doesn’t seem to work in every case. There is a clear difficulty here when the two conditions on rational belief: Consistency (which we might call the external condition) and evidential support, (the internal condition) pull us in different directions. 7 This procedure was intended to apply to classical inference, but the method obviously generalizes to cover cases in which the base inference is X.

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95

In the lottery paradox, for instance, we seem to have good evidence for each one of the lottery beliefs (ticket 1 won’t win, ticket 2 won’t win,. . . , ticket n won’t win.) and we can make the evidence as strong as we like by making the lottery ever larger. Now conjoin the beliefs and we get ‘No ticket will win.’ which contradicts fairness. We could get consistency by throwing out the belief that the lottery is fair, but that would be cheating. The problem is that each of the lottery beliefs has exactly the same support as the others. They stand or fall as one, it would seem. If we let them all fall, then the rationality,or at least the non-irrationality of buying a lottery ticket would seem to follow. But isn’t it true that it isn’t rational, according to the accepted canons at least, to buy a lottery ticket?8 Leaving aside the possibly controversial issue of the lottery paradox, take any situation in which we are unable to find a rationale for discarding one member of an inconsistent subset rather than another. Quine seems to suggest that in this situation, the counsel of prudence is to wait until we do find some way to distinguish among the problematic beliefs. Those with less patience seem to regard random discarding until at last we get to consistency, to be the path of wisdom.9 We are inclined to reply to Quine that patience, for all that it is a virtue, is sometimes also a luxury we cannot afford or even a self-indulgence that we do well to deny ourselves. To the others we say consistency is not a virtue which trumps everything else. Suppose we might achieve consistency by throwing away one of ˛ or ˇ though we have no reason to prefer one over the other. Flipping a coin is a method for determining which goes to the wall, but we have no way of knowing if we have determined the correct one. We have left ourselves open to having rejected a truth and accepted a falsehood. ‘Yes, but at least we now have consistency!’ won’t comfort us much if the consequences of picking the wrong thing to throw away are unpleasant enough. Let us take up level once more.10 In saying the level of the set † is 8 Not

for nothing have lotteries long been known as ‘a tax on fools.’ seems to be the route advocated by some of those in computing science who have devised so-called truth-maintenance systems. 10 We realize that the Quinean suggestion is not the only one, though it might be the most 9 This

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k, we are saying two things. First that there is a way to divide the logical resources of † into k distinct subsets each of which is consistent. From now on we shall refer to these consistent subsets as cells. Second, that any way of thus dividing † must have at least k cells. Here we have got to the level of consistency not once, but k times. Not only might we wonder which of the k cells is the ‘real’ one, the one which best represents ‘the way things really are,’ but there may be lots and lots of distinct ways to form the k cells. Which of the possibly many ways should we privilege? At this point, we cannot answer these questions, which means that we must treat the cells on an equal footing along with the various ways of producing them.11 In saying this, we say that we shall count as a consequence of † in the derived inference relation, whatever formula follows (in the ‘underlying’ logic, say X) from at least one cell in every way of dividing † into k cells. When the underlying logic is X , the derived inference relation is  called X -level forcing, which relation in indicated by X . We can give the precise definition as: Definition 6.  € X ˛ if and only if, for every division of € into `X .€/ cells, for at least one of the cells ;  `X ˛ It is easy to see that:  Proposition 3. If € X ˛ then `X .€/ D `X .€ [ f˛g/ Proof. Suppose the condition obtains and let `X .€/ D k. It follows from the definition that every division of € into k cells, results in at least one cell that X-proves ˛. But then we could add ˛ to the cell in question without losing the cell property since X is a logic which preserves consistency. In such a case, after adding ˛ we would have a well-known, to deal with inconsistent sets of formulas. We do not, however, intend this essay as a survey of all of so-called paraconsistent logic. 11 Which is not to say that there is no way. Elsewhere one might find suggestions which narrow the range of ways of dividing up our initial set. In this connection see the discussion of A-forcing in chapter 9.

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division of € [ f˛g into k cells. Moreover there couldn’t be a division of € [ f˛g into fewer than k cells without there being a similar division of € which would contradict the hypothesis.  It obviously follows directly from this that:  Corollary 1. X preserves X-level, in the sense that `X .€/ D `X .Œ X .€//

6.6 Yes, But Is It Inference? To be perfectly honest, or at least honest enough for practical purposes, all we have shown is that X-level forcing is a relation that preserves  X -level. There is a gulf between this, and the assertion that X is an inference relation which preserves X-level. The obvious problem for anybody wishing to assert such a thing resides in the fact that we haven’t, for all our efforts at precision, actually said which relations count as inference relations. What we have said, is that we assume that the inference relations we mention admit certain rules. Shall we take the collection of these rules to be constitutive of inference? We shall not, because some, at least, of the rules which the underlying logic admits simply don’t make sense for the derived relation. This should not come as a surprise. It is our palpable annoyance with the un derlying logic which leads us to propose X . How silly then to require that the derived logic inherit everything from the underlying logic, since that would make the derived logic another source of irritation rather than the balm for which we hope.  Although it is easy to check that X inherits from its underlying inference relation X both [R] and [Cut], we can see that it fails to admit the rule [M] of monotonicity, which we would do better to label the rule of unrestricted monotonicity from now on. But this is one of those cases in which the rule ought not to apply to the derived relation. If we are allowed to dilute premise sets in an arbitrary way, there is nothing to prevent us from raising the X -level of such sets. But raising the level gives us, in general, (logically) weaker cells in each logical cover. What used to X-follow from at least one cell of every such cover might no

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longer do so, as we are cut off from vital logical resources by the finer division.12 This is not to say that no form of monotonicity makes sense for the derived relation. Quite the contrary, in fact, what most emphatically does make sense is that X-level forcing consequence must survive any dilution which preserves the level of the premise set. Such a restricted version of monotonicity manifestly is a rule for X-level forcing, as is trivial to verify. Along with level-preserving dilution, there are certain consequences which must survive any dilution at all, whether or not the X-level of the premise set increases. These are the consequences which dilution cannot affect, and we can say precisely which they are: The X -consequences of the empty set and of any unit set will remain X-consequences of at least one cell of every logical cover of any set which contains any of these privileged sets. In earlier work these sets were called singular. The other properties which we have been mentioning for the underlying logics are non-triviality and having denial. It should be clear that when the underlying logic is non-trivial so will be its derived forcing relation. In fact, keeping to our original definition of consistency, in passing from an underlying logic X to its derived X , many of the  X -inconsistentsets fail to be X -inconsistent, which is after all, the whole point of the derived relation. Which brings us to denial. If the underlying logic has denial, nothing follows about the derived forcing relation, which is not necessarily a bad thing. This is because in the underlying logic, inconsistent sets are (typically) relatively easy to come by, but in the derived logic, the only inconsistent sets have inconsistent unit subsets, or what we called X -absurd formulas. Having denial doesn’t imply having absurdities. So the derived logic will have denial only if the underlying logic has absurd formulas, but in no case will the derived logic have non-trivial denial. And since ˇ denies ˛ in the derived logic if and only if one or both 12 Here is a concrete example where the underlying logic is classical. The premise set f˛; ˛  ˇ g has the classical-level forcing consequence ˇ since it has level 1. If we add the formula :ˇ the resulting set has level 2, and now there is a logical cover: f¿; f˛; :ˇ g ; f˛  ˇ gg no cell of which classically proves ˇ . Thus ˇ is not a classical-level forcing consequence of the diluted set.

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of the two are absurd in the underlying logic, the principle [Den] must hold of the derived relation, but it is much less interesting there than it is in the underlying logic. So for the derived relation, we would seem to be on solid ground when we require [R], [Cut], and the restricted version of monotonicity. Those we might well regard as the hallmarks of inference, or at least of derived inference. And let us not forget that the derived relation agrees exactly with the underlying relation X on the consequences of the X consistent sets. So for this reason alone, we ought to admit the forcing relation into the fold. Perhaps we should put it this way: Anybody who thinks that X is fine and dandy except for its failure to be properly sensitive to the varieties of inconsistent sets, must think that X -level forcing is an adequate account of inference. This is because when premise sets are X consistent, X-level forcing just is X . And while it surely isn’t X for (some) X -inconsistent sets, in those cases X isn’t an inference relation. X has abdicated, throwing up its hands and retiring from the inferential struggle, offering the hopeful reasoner nothing beyond a contemptuous ‘Whatever!’

6.7

Forcing In Comparison With Other Level-preserving Relations

Finally, we consider the place of the X -level forcing relation compared with other possible relations which preserve X-level. We cannot claim uniqueness here, for there may be plenty of relations, even inference relations, which preserve X -level. What we can claim however, is inclusiveness, in a sense to be made precise. That precision will require another property13 of the underlying logic X.

13 If we regard deductive systems as categories, then to call a logic productival is simply to say that the category (logic) X has products. The first condition amounts to the assertion of canonical projections while the second amounts to the universal mapping property of products.

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Definition 7. A logic X will be said to be productival if and only if for every finite set € there is some formula  such that  `X for every 2 €, and € `X 

Evidently being productival is another of those properties more honored at the level of underlying logics. If a productival logic X has denial, then X-level forcing will certainly not be productival. But of course at the underlying level, products are useful. For instance: Theorem 3. If X is productival then for any pair €; ˛ with € a finite set of formulas and ˛ a formula: If Y preserves X-level and admits level-preserving monotonicity, then  € `Y ˛ H) € X ˛ Proof. We shall content ourselves with a sketch only—a fuller treatment can be found in ‘Level Compactness’ (Payette and d’Entremont, 2006). Assume for indirect proof that € `Y ˛ and that € fails to X-level force ˛. From the latter we know that € has finite level, say k, and that there is a logical cover of € of width k such that none of the k cells X-proves ˛. Where ˇ (non-trivially) denies ˛ add ˇ to each cell and then form each of the k products of the cells. The set of these products must have X -level k but the Y closure of the set must have X-level k C 1. So Y fails to preserve X-level, contrary to hypothesis.  The restriction to finite premise sets will chafe us only until we see its removal in the more general result referenced above. So while there may be many inference relations which preserve Xlevel, X -level forcing is the largest of them.

Seven

Preserving Logical Structure G ILLMAN PAYETTE

Abstract In this paper Gillman Payette looks at various structural properties of the underlying logic X, and ascertains if these properties will hold of the forcing relation based on X. The structural properties are those that do not deal with particular connectives directly. These properties include the structural rules of inference, compactness, and compositionality among others. The presentation of the logic X is carried out in the style of algebraic logic; thus, a description of the resulting “forcing algebras” is given.

7.1

Introduction

Faced with the possibility of logical pluralism, if not the reality of it, one should follow a methodology of research which is sensitive to the plurality of logics. What has not been dealt with so sensitively is the question of how best to deal with inconsistent sets if we don’t want to trivialize inference in every one of these cases. In classical and intuitionistic logic the sets fP ^ :P g and fP; :P g have the same deductive closure: Everything! In Latin the rule of infer101

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ence is phrased as ex falso quodlibet, which may be translated as ‘from the false whatever.’ This is the notion of inconsistency first presented by Post and used throughout this volume and elsewhere. For sets of formulas this general definition of inconsistency has it that a set is inconsistent just when everything follows from it.1 It should not escape our keen attention that the definition is mute on the subject of falsity in general, and that there is likewise no mention in particular of the negation connective. The paraconsistent 3-valued logic of Priest, the Logic of Paradox (LP), uses the same language as classical logic, and has all of the same theorems.2 Thus, it has .P ^ :P /  Q as a theorem so, in a sense, it rings true that from the necessarily false, i.e., the absurd, everything follows in LP. However, the two sets above do not have trivial closures in LP. So, this logic can, in its way, deal with inconsistency. But, falsity and truth apply to sentences. Only by type raising do they apply to sets. What has happened, historically speaking, is a running together of absurdity, inconsistency and falsity. LP still runs together absurdity and inconsistency; it merely renders the two, inert. Although the sets above have the same closure in classical logic (and LP respectively) there is an obvious difference in the composition of the two: one contains what is often taken to be an absurd formula, and the other does not. One might ask ‘Is there a logic that can distinguish between these two kinds of set?’ But, this question may be wrongheaded. Inconsistency, as well as consistency, is a relative notion; it is relative to a particular logic. Therefore, when a logic like LP says that fP; :P g does not explode it says that this set is not LP-inconsistent. Lacking another notion of inconsistency, we see that some paraconsistent logics merely regard what are inconsistent sets in other logics as consistent. This shifts the ground by proposing what was called in Chapter 2 a replacement of the original theory (logic), but it leaves the originating problem untouched. Think of it this way: Axiomatic set theory does not fix naive set theory—the latter remains paradoxical. Neither does LP fix the classical account of inconsistency—we are still unable, classically, 1 Such 2 See

sets are often said to explode (inferentially). Avron (1994).

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to distinguish inferentially between classically inconsistent sets which contain an absurd formula and those which do not. In opting for a replacement strategy, what happens is that the rules of the game have changed to solve the problem. Of course this may be entirely respectable—it may even be the only way to proceed. We change the rules all of the time. But, when the rules have changed in logic it seems that the meanings of the connectives involved change as well. And that may be a problem. Consider a set that we regard as consistent. In that case the logic does not permit us to treat that set as classically consistent. We must continue according to the rules unless we are constantly changing the meanings of the connectives to suit our desires. But that seems wrong. A different solution is possible. There are many logics which suffer under the iron yoke of ex falso. Thus, it would make sense to develop, rather than a new logic to deal with classical account of inconsistency, a method of determining, for a whole class of logics which have this problem, a notion of derived consequence that enables the distinction we want without having to construct a whole new logic in each case. We seek a method which we can apply to a logic X described quite generally so long as we can represent X’s notion of inconsistency as the explosive Postian kind. This may be considered a method of Paraconsistentizing the logic.3 The notion of Forcing a` la Schotch and Jennings (1989), which is the subject of this volume, is what shall be used, but massaged into the framework of algebraic logic. Just what constitutes a good way of constructing such a derived inference relation is obviously crucial. The Quinean idea of minimal mutilation is what we take to be an obvious criterion for ‘goodness.’ The more of a logic that can be kept the better. What is of interest here is how the ‘structural properties’  of an underlying logic X may be transferred to the forcing relation X . We call properties ‘structural’ when they don’t have a direct relation to particular connectives. These are properties like compactness, preservation of consequence by substitutions, and the structural rules of inference. The compositionality of a logic, 3 This term was used first by Alexandre Costa-Leite; however, the method that appears here is different from his approach.

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i.e., the meaning of whole is given by the meaning of the parts, is also a structural property since it does not deal with particular connectives. I will also make a few comments on how the existence of certain kinds of formulas, like denials,4 may affect these structural properties. The present study makes up part of the justification of the forcing relation as an acceptable notion of paraconsistent consequence. The point is to see that much of the general structure of a logic X can be preserved by the forcing relation. The other part involves the idea of level preservation which is taken up in detail in Chapter 6 of this Volume. We also show where the forcing relation sits amongst other similar relations on X. The final discussion concerns first-order classical logic as an instance of many of the properties mentioned above.

7.2

Definitions

This study is diverse and long. Therefore I will present many definitions to give some background to what follows. In what follows there is a a minor departure from previous work this will be pointed out as it is necessary. The reason for the departure is the focus on the semantic version of a logic, rather than the syntactic. First we will begin with a logic. Logics Definition 1. A logic X is given by a tuple hSX ; `X ; MX ; ˆX i such that 1. SX is the set of formulas. 2. `X is the provability relation, thus proof theoretic consequence. This may not exist for some logics. 3. MX is the class of models. Whatever it may be. 4. ˆX is the satisfaction relation between models and formulas. 4 in

the sense of that term introduced in Chapter 6 of this volume

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Note that ˆ is not semantic consequence. Semantic consequence, or entailment, is defined in the usual way using ˆ, i.e., €  ˛ iff for all M 2 M; M ˆ € H) M ˆ ˛. I will follow the convention and use the  symbol for both relations. Also, when M  ' for every M 2 M we write  ' which means ' is a truth of the logic. Relative to each logic there is a consistency predicate CONX which holds € when either €  ˛ for every ˛, or, there are no models of €. An important part of the algebraic presentation of logic is the meaning function mng. The meaning function will, in a sense, be definitive of the logic since the meaning algebra of a logic is what is studied in algebraic logic. We will present a candidate for the meaning function of the derived forcing relation, and show some properties which it inherits from the meaning function for the original logic. So in what follows a logic is given with a meaning function right from the start. Definition 2. A meaning function for a logic X is a map mng W MX  SX ! H where H is some class. Thus it takes formula-model pairs to something. A function mng is a meaning function for a logic X –subscripts omitted–just when .8M 2 M ; ' 2 S/Œmng.M; / D mng.M; '/ H) M ˆ ' ” M ˆ . There could be many meaning functions for a logic but, there are canonical examples. The composition of the set H is left undefined in general; it will vary from logic to logic as the composition of M does. In the case of first-order logics the meaning of a formula, relative to a model, is given as the set of variable assignments that satisfy that formula in that model. For the usual 2 valued propositional logic the models are just the truth value assignments and the meaning functions are their extensions to all of the formulas. A meaning function will only be of use if its image forms an algebra. An algebra is a set A (often called the underlying set of the algebra) along with a set of functions from An to A for various natural numbers n (including zero). A logic is said to ‘have connectives’ when for any n-tuple of formulas 1 ; :::; n and any n-ary connective 4 in the set Cn

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of connectives of the language of X, the formula 4. 1 ; :::; n / is also part of the language. The set of formulas then can be seen as an algebra, ˚ X . Recursively genand it is called the ‘formula algebra’ denoted by A erated sets of formulas will produce formula algebras. In that case, the underlying set is just the set of all formulas generated, and the functions are the connectives. A function (or map) h W hA; A i ! hB; B i is a homomorphism when h.a A b/ D h.a/ B h.b/, and if h W A ! A, then h is an endomorphism. Homomorphisms are structure preserving maps. When the meaning function of a logic is a homomorphism for each model, the logic is called compositional. We can give a nice mathematical definition of compositionality as follows. The subscripts for the logic X are omitted where confusion is unlikely. Definition 3. A logic X is compositional just when the meanings of compound formulas depends on the meanings of the component formulas. More precisely, let 1 ; :::; n and '1 ; :::; 'n be n-tuples of formulas from S, and 4 an n-ary connective in the language of X, then given a model M 2 M if for each 1  i  n mngM . i / D mngM .'i / then mngM .4.'1 ; :::; 'n // D mngM .4.

1 ; :::;

n //

When a logic’s meaning function is a homomorphism for each model we can generate a meaning algebra for a logic. That is what it means for the image of a meaning function to form an algebra. There are two interesting sets that deserve mention. Definition 4. Let K  M and †  S. 1. The ‘theory of K’ is T h.K/ D f 2 S W M ˆ 8M 2 Kg , i.e. the set of formulas every model in K models.

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107

2. The ‘models of †’ is the class Mod.†/ D fM 2 M W M ˆ 8 2 †g. Which is the class of models which model all the formulas in †.

The aim of this essay is to discuss the properties which carry over to the forcing relation. In furtherance of this aim we introduce some important properties of consequence relations. The substitution property comes in two flavors viz. weak and strong. A substitution s is a morphism from the set of atomic formulas to the set of all formulas which can be extended uniquely to a homomorphic endomorphism of the formula algebra. So a substitution s on a propositional language maps each atom to a formula, not necessarily atomic. Then one can reconstruct formulas with the substituted atoms. Definition 5. The substitution properties for a logic X are: 1. (Weak) If ˆ then ˆ .P =s.P // where .P =s.P // is the uniform substitution of s.P / for P in , with P an atom and s a substitution. 2. (Strong) If €  , then € s  s where € s is f' s W ' 2 €g and ' s is '.P =s.P // for each atom mentioned in '.

A logic which obeys the substitution properties preserves consequences under uniform substitutions of the atoms in the formulas. The strong property implies the weak property by letting € D ¿. The weak property asserts that theorem-hood or truth-hood is preserved by uniform substitutions. There are logics that have what are called ‘structural rules’ of inference. Such logics are not called ‘structural logics’; however, every structural logic must have these rules. A common presentation of the structural rules is (I omit the subscripts on the `):  [R] for reflexivity, if

2 € then € `

.

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 [M] for monotonicity, if € `  [Cut-1] if € `

and €;

and €   then  `

and

` ' then € ` '.

The first two rules are fine; however, there are many formulations of [Cut]. The formulation we shall use is the following. [Cut]: Let € ` ı for all ı 2 0   and  ` then €;  0 ` . Where  0 is set of members of  not in 0 . This version of the rule is slightly more general than the one above. It is also more general than the following version: [Cut-2]€ ` ı for all ı 2  and  ` ˛ then € ` ˛. But the previous version is a bit too strong for our purposes in this paper thus we use [Cut]. There is a very interesting fact to be notice about the last formulation of the rule. Proposition 1. [R] and [Cut] imply [M]. Thus, only in certain situations will the three rules really ‘come apart’; forcing is, as will be shown, one of those situations. Definition 6. A logic X is structural if and only if it obeys both the strong substitution property (see below) and the structural rules of inference (with [Cut-2] in place of [Cut]).

Covers, Models and Levels Essential to forcing are the notions of cover and level. A cover C in X, or an X-cover, is a tuple of sets of formulas indexed by some ordinal  C 1  ! C 1: C D hi W i 2  C 1 & 0 D ¿i such that for each i 2  C 1, CONX .i / where CONX is the consistency predicate for X. The  from the index ordinal of C is referred to as the width of the cover C and denoted w.C/. Each i in C, indicated by a pardonable abuse of notation as ‘i 2

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109

C’, is called a cell. The collection of X-covers will be referred to as M  . This notion of cover departs from the definitions in the literature, e.g., Schotch and Jennings (1989); Brown and Schotch (1999) and the present volume, but it will help avoid technical problems later. Since a cover is defined as an ordered tuple of length less than or equal to ! 5 the condition for two covers C; C to be equal is: w.C/ D w.C / D  and 8i   i D i , where i 2 C and i 2 C . So covers may have repetitions of cells, or two covers with different orderings of the same sets are not equal. Definition 7. C is a cover of €, that is C  €/, just when 1. 0 D ¿ 2.

CON X .i /,

8i 2  C 1

3. For each 2 € there is a i 2 C such that i  and, 4. if the constant symbol a or free variable x of the language of X is mentioned in i for some i 2  C 1 then it is mentioned in €.

For a symbol t to be mentioned in a set means that for some formula in the set t appears in that formula. It is easy enough to see that a cover covers a single formula when there is a cell in the cover that entails that formula. In such case it is written C  . The condition on terms and constant symbols will be of use when we discuss first order logic. As the definition stands it is only a small divergence from the one found in earlier literature. This notion of cover can be interpreted as something like a model; it is a kind of model for certain inconsistent sets relative to the logic X , though not all X -inconsistent sets will have these kinds of models. To see that the C’s function as models in this way suppose C  . Thus, there is some cell i in C that X-entails , and CONX .i / by 5!

is the first countable ordinal, or the order type of the natural numbers.

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the definition of cover. We call a formula absurd when its unit set is inconsistent. An absurd formula does not have any models, nor does it have any covers. Consider an absurdity ˛. If ˛ had a cover, then there would be a C and i 2 C such that i  ˛, but that would require that CON X .i /, because X has [Cut], which contradicts the second condition in the definition cover. Thus, sets like f˛g have no covers and the converse of this also holds. Note the important assumption that the logic X is not trivial, i.e., the empty set’s closure is not everything. Proposition 2. € does not have a cover iff € contains an absurdity. A related fact is that the minimal cover for a set is a cover just when there are covers for a set. The minimal cover of € is Cm € D m m h¿; f g W 2 €i. C€ is minimal in two senses. The cover C€ is the finest partition one can make of the set €, and it is ‘contained’ in any other cover of the set. It is contained in the sense that, for any cover C   of a set €, if Cm €  ˛ then C  ˛. This cover will be of use when discussing infinite levels. A level function relative to X, `X , is defined using the notion of cover. The codomain, or range, of the level function will depend on the size of S. We restrict the cardinality of S to the countable cardinal for this essay; thus, jSj D !. Definition 8. The X -level (i.e. level relative to the logic X) of the set € of formulas of the underlying language of X, indicated by `X .€/ is defined: 8 < min ŒCOVX .C; €/ if this limit exists `X .€/ D w.C/ :1 otherwise

Thus, the level of a set is determined by the minimum width of the collection of covers of a set €. This is of course a function, and under the assumptions made in this paper its domain is ! [ f!g [ f1g. Level actually induces a kind of measure on the set €. Level measures how

7.2. DEFINITIONS

111

inconsistent a set is. One must use more classically consistent sets to cover fP ^ Q; :P ^ R; :R ^ :Qg than to cover fP; :P; Q; Rg. And one cannot cover fP ^ :P g; it has level 1! This is how we can finally see the difference between inconsistency and absurdity. An inconsistent set may have a level, but absurdity does not. Also note that if there are theorems, then any set of theorems has level 0. Since the cover C D h¿i is a cover for a set of theorems, w.C/ D 0 because C is indexed over the ordinal 1=0+1= C 1. With level we can define the forcing inference relation on P .S/  S.  Definition 9. [Forcing]6 € X € & w.C/ D `X .€// H) C 

” .8C/Œ.C  .

This is read ‘€ level forces , relative to X ’. A set will force a conclusion, , just when every cover of € of width ‘level of €’ is a cover of . This relation in some cases will collapse into just what follows from the unit sets, but not always, and, considering the size of P .S/, one may say only half of the time. What is needed to fill out the algebraic approach to forcing is a meaning function. However, meaning functions are semantic entities; they take models as one input and formulas as the other and spit out something. If the derived meaning function for forcing is to depend on the underlying logic it will, ideally, depend on the models of the underlying logic. The models for the forcing relation are, as mentioned above, covers. Thus, what must be constructed are covers, not out of sets, but, rather, out of models. This can be accomplished by the following construction. Definition 10. A semantic cover F of a set €, written F  €, is a tuple of models from M , hMi W i 2 i,  an ordinal less than ! C 1, such that for any 2 € there is an Mi 2 F such that Mi  .

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Thus, semantic covers are just tuples of models. Again, the  in the definition above will be referred to as the width of the cover, and covering a single formula is to cover its unit set. In this case the width of the cover is just the cardinality of the cover. The level of a set of formulas is defined thus, 8 ˆ 0 if €  CX .¿/ ˆ <  `X .€/ D min ŒF  € if this limit exists w.F/ ˆ ˆ : 1 otherwise where CX is the consequence closure operator for X. It is the set of all consequences of a set of formulas relative to the logic X. Of course a forcing relation can then be defined relative to this notion of cover in an analogous way. Definition 11. €Œ ˛ if and only if for all F  € such that w.F/ D `X .€/, F  ˛. But, do these two notions of cover give rise to the same forcing relation, extensionally speaking? The sort answer is yes. Lemma 1. If C is a (syntactic) cover of € then there is a semantic cover of € of the same width, and vice versa. Proof. Suppose that C  €, and w.C/ D . Then for any 2 € there is i 2 C such that i  . Thus if M 2 Mod.i / then M  . Each Mod.i / ¤ ¿ by definition of cover, so let FC D hMi W some Mi 2 Mod.i / i ¤ 0i. So the width of F is that of C, and every member of € is modeled by some member of F. The opposite direction follows by making up CF of ¿ and i D T h.Mi /, Mi 2 F.  And what about the different types of level? Lemma 2. For all €  S, `X .€/ D `X .€/.

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113

Proof. Suppose that `X .€/ D 0. Obviously `X .€/ D 0, and vice versa. If `X .€/ D , then let C  € and w.C/ D . If the levels were different, then we could generate a semantic cover, or regular cover, with the Mod and T h operators respectively to derive a contradiction. If `X .€/ D 1 Then there are no covers of € of either kind, because € must contain an absurdity. This is the case with either definition of cover.  Then we can prove the result that we want.  Theorem 1. For any € and ˛, € ˛ ” €Œ ˛. Proof. Suppose €Œ6 ˛. Then suppose C 6 ˛. Construct a semantic cover out of C as in lemma 1. By lemma 2 this cover has the same width, which is the `X .€/, as C and it is a semantic cover of €. So €Œ6 ˛. For the other direction, the dual argument as in lemma 1 is used and the definition of the T h operator.  With the two notions amounting to the same thing from two different directions I will use whichever notion is appropriate for the context.

7.3

Substitutions

Recall definition 5. There are two versions of a logic being syntactically substitutional. One is strong the other weak. Given that X is structural, the forcing relation is weakly substitutional (proof omitted), and only obeys a qualified version of the strong substitutional property.7 I shall prove the theorem that forcing has a restricted form of strong substitution. Recall that the logics considered are ones that obey the three structural rules mentioned in the last section. In what follows I will use the syntactic version of forcing since I am discussing a syntactic notion of substitution.

7 An interesting side note to this point is that forcing will also have the interpolation property if the underlying logic does.

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Definition 12. A ‘Partition S Cover’ of € is a cover C such that w.C/   D `X .€/ and C D €. Further, if i and k .i; k  / are distinct members of C then i \ k D ¿. This definition says that C is a partition of the set €. It is the kind of partition in which each cell is a consistent set relative to X. Such a thing is not unique in general, but there is always the possibility of having such a partition if the set’s level is not 1. Lemma 3 (Partition Cover Lemma). If `.€/ D  2 !, then, given a cover C of €, there is a partition cover C0 which one can construct from C. Proof. Assume that `.€/ D . Then let C be a cover whose width is . We shall construct the partition cover out of this one. First, let 0i D CX .i / \ € for all 1  i  . Note that i does not start at 0. Once the construction has finished we add ¿ to the rest of the sets to get the C0 wanted. Let ij D 0i \ j0 where i ¤ j . Further, define k D

k[1

.0k

.0k \ 0m //

mD1

for k  . Since C is well ordered by its width ordinal one can speak of the ‘first i such that...’ . In a sense the  ’s check to see if they are disjoint with all of the sets ‘before’ them. S Claim: These k ’s are all disjoint, and € D k k . Suppose i ¤ j , and 2 i \ j . Without loss of generality assume that i < j , since it is one or the other. Then, since 2 i , it must be that 62 0k for all k < i , but 2 0i . So, 2 0i \ j0 . However, by the construction of jS it is impossible that 2 i , since i < j . It is clear that k k  € so assume that 2 €. Then for 0 some i 2 C, 2 CX .i /, but that means 2 i . We may assume that i is the smallest such index, since there must be one. If 62 i then there would have to be S a smaller j such that 2 j0 which S is not  possible. Thus, 2 i  k k  €; therefore, € D k k .

7.3. SUBSTITUTIONS

115

Let C D h¿; i W 1  i  i. Then C is clearly a partition cover of €.  The next lemma allows us to restrict what covers we must consider of the collection of covers of a set €. Notice that although this lemma is proved for the forcing case it will work in the X  case since in the latter case all covers are considered. Only covers whose width is the level of the set € are considered in forcing. Thus the obvious extension will suffice for that case.  Lemma 4. For `X .€/ > 0, € ” 8 C a partition cover of € of width `X .€/, C  . Proof. ( H) ) If €Œ then all covers of € of appropriate width will cover . Partitions are covers of €, so they must cover . ((H) By contrapositive. If €Œ6 then there is a cover C of €, of width `X .€/ which does not cover . Let C be a partition cover of € associated with C as in the partition cover lemma, lemma 3. Notice that i  CX .i / for i 2 C. This is because i  0i , and 0i D CX .i / \ €. Thus, CX .i /  CX .i / by assumptions on X which transfer to CX . So, 62 CX .i / for all i 2 C, and 62 CX .i / for each i 2 C . Thus, C 6 .  The case where `X .€/ D 0 and € is non-empty will not have any partition covers of width 0 since the only cover of width 0 is h¿i. However, this poses no problem because if a set of theorems forces something then that something is also a theorem. But forcing is weakly substitutional. We can see that strong substitution does not hold for forcing by considering the following counterexample. Let € D fP; :Qg. Let the substitution s be the function which takes P 7! P and Q 7! P . As we shall see in the next section forcing is just like the underlying logic X when the sets are consistent; thus, €Œ P ^ :Q. Notice though € s D fP; :P g and s.P ^ :Q/ D P ^ :P . But then € S Œ6 P ^ :P since forcing is level preserving (see the next section theorem 6). Although the underlying logic may be substitutional, the forcing relation is not.

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Having said this if we restrict the class of substitutions, we can derive a substitution theorem for substitutions from that class. Theorem 2. Suppose that s is a substitution from the set of atoms ˚ to the set of all formulas such that s 2 g 2 SA W `X .€ g / D `X .€/ , i.e., all of the substitutions which are level preserving. If X is such that for 0 0 any pair h€; 'i and any substitution s 0 , €  ' H) € s  ' s ; then €Œ ' H) € s Œ ' s . Proof. By contraposition. Assume that for some level preserving substitution s, € s Œ6 s . X has been assumed to obey the substitution property which is equivalent to: if s 6 ' s then  6 '. Let C be a partition of € s such that C 6 , which exists by definition. Then for each i 2 C, i 6 s . Consider the sets s 1 Œi : the inverse image of the i ’s from C. If s is applied to these sets the i is returned, since each member of the inverse image gets mapped to some element of i , and since i  € s , there must be something which was mapped to each element of i . Thus, .s 1 Œi /s D i . Let i D s 1 Œi  \ €. Claim: i 6 . Suppose that i  . Then, by the assumption that X has the substitution property, .i /s  s . But i D .i /s . So i  s , but it does not. Thus, i 6 . This implies CONX .i / for all i 2 C. S Claim: €  i . Each element of the i ’s is an image of some element of €, so all of € is recovered in the inverse images of the i ’s. We don’t get more than € since we restrict the inverse images to € in the construction of the i ’s. Let C D h¿; i W i 2 Ci. This is clearly a cover of €, and not a cover of . But there are only as many i ’s as there are i ’s; thus, C has the same width as C. But the width of C is the level of € s , i.e., is equal to `X .€/. So €Œ6 which is what is wanted.  Thus not all is lost, but there must be some room given for failures to balance out the gains.

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117

7.4 Algebraic Concerns There are two types of algebras which are of particular interest to the algebraic logician. These algebras arise from a logic through the application of the meaning function. The first type of algebra is the meaning algebra, and the second is the Lindenbaum-Tarski algebra. The meaning algebra is constructed out of the meanings of the formulas of the logic. Lindenbaum-Tarski algebras are made up of equivalence classes of formulas. Both are necessary to study the properties of the logic in question. However, forcing has many of the necessary properties that make it susceptible to algebraic methods, but it fails in a crucial place. To begin this study some definitions are needed. Definition 13. [Algebra, subuniverse] An algebra A D hA; A i is a set Anwith a collection of o (finitary) operators A D fik W i 2 ; k 2  . The  is referred to as the ‘type’ of the algebra and A as the ‘universe’ or ‘carrier’ of the algebra. A subuniverse B of an algebra A is a subset B of A such that B is closed under the operations in A . The important definitions for algebraic logic are as follows: Definition 14. A Lindenbaum-Tarski (LT) algebra is given with respect to a class of models K. The LT-algebra is a isomorphic8 ˚X copy of copy of the quotient algebra of the formula algebra A modulo the equivalence relation K : K ' iff mngM . / D mngM .'/ for all M 2 K. The class of LT-algebras is denoted AlgLT .L/. Definition 15. The meaning algebra relative to a model M is denoted mng.M/. It is the set fmngM .'/ W ' 2 Sg. The type of mng.M/ is the interpretation of the connectives of the language

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in the models of X. The class of meaning algebras for L is denoted as Algm .L/ With these definitions in place it is time to construct meaning functions for forcing. Meaning Functions and X  Meaning functions have less to do with the consequence relation and more with the relation between the class of models and formulas. For the rest of this section we will assume that M is a set. The idea now is to, somehow, define a meaning function for X  from the meaning function for the logic X. The one presented may not be the only one available, but it does do enough of the work we would want it to. In this section I will use the semantic version of cover. Recall that the “models” of X  are (semantic) covers so the meaning function will have to be a function which takes cover-formula pairs to some class. Let F D hMi W i 2 i 2 M  and ' 2 S. Then, mngF .'/ D hmngMi .'/ W Mi 2 Fi So for two instances of mngF to be equal, at and ' say, means that for each Mi 2 F, mngMi .'/ D mngM function Si . /.The meaning   that has been defined maps M  S to 2.!C1 ¿/ .H / , i.e., each mngF .'/ 2 .H / where  is the non-zero width of F. Before proceeding it must be proved that this mng  is a meaning function. The reader should recall definition 2. Theorem 3. mng  is a meaning function for X  . That is, for ; ' 2 S ” F  '. and any cover F, if mngF . / D mngF .'/ then F  Proof. Let mngF . / D mngF .'/, for some ; ', and F. Suppose F  '. By definition there is some Mi 2 F such that Mi  '. By hypothesis, mngMi . / D mngMi .'/. Since mng is a meaning function for X , Mj  ” Mj  ' for all Mj 2 F, i.e., it  does so for Mi . Thus, F  . And similarly from the assumption that C  . 

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119

Thus mng  is a meaning function. What can be said about compositionality? Theorem 4. If a logic X is compositional, then the derived logic X  will also be compositional relative to the meaning function mng  . Proof. Suppose that X is compositional. Then suppose there is an n-ary connective 4, and some n-tuples of formulas as in definition 3 such that given a cover F 2 M  , for each 1  i  n, mngF . i / D mngF .'i /. These assumptions imply that given a Mi 2 C and j and 'j one has mngMi .

j/

D mngMi .'j /

This implies that one has mngMi . 1 / D mngMi .'1 /; :::; mngMi . n / D mngMi .'n /. By hypothesis mng is a compositional meaning function so it follows that: mngMi .4.

1 ; :::;

For each Mi . So, mngF .4. is what is wanted.

n // 1 ; :::;

D mngMi .4.'1 ; :::; 'n // n //

D mngF .4.'1 ; :::; 'n //, which 

Algebras What can this information do for this study? It is clear that a meaning algebra can be formed relative to any semantic cover F. But one can also form the quotient algebras relative to covers as well. So, one can generate both LT- and meaning algebras. Indeed, it is possible, but what can be said about the resulting algebras given knowledge of the underlying algebras? Actually, more than one may expect. In the case of meaning algebras, which is the more complex case, the meaning algebra relative to a cover is a subsuniverse of a product of meaning algebras. Recall that mngF .˛/ is a tuple of meanings of the underlying logic. That means it is a member of the direct product of the meaning algebras in F. That is, Y mngF .˛/ 2 mng.Mi / Mi 2F

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However, it is not clear that mng  .F/ is the product. Think of the product displayed as an array with the formulas heading the columns, and the models from F heading the rows. mng  .F/ is made up of just the columns, no other combinations of elements from the meaning algebras appear. ˛

M1 M2 M3 :: :

ˇ

ı

::: mngM1 .˛/ mngM1 .ˇ/ mngM1 . / mngM1 .ı/ : : : mngM2 .˛/ mngM2 .ˇ/ mngM2 . / mngM2 .ı/ : : : mngM3 .˛/ mngM3 .ˇ/ mngM3 . / mngM3 .ı/ : : : :: :: :: :: :: : : : : :

But, because of compositionality, these columns are closed under the operations of the product algebra. Thus, mng  .F/ is a subset of the product which is closed under the operations on the product, i.e., a subsuniverse. Of course, if one knows more about the class of meaning algebras of X then one can say more about the meaning algebras of X  . For instance it is well known that the class of meaning algebras of classical propositional logic is a variety. This means that it is closed under products and subalgebras, among other operations on the class. Thus, for such logics, the forcing meaning algebra is a member of meaning algebras for the original logic. Consider the LT-algebras. These are made of quotients of the formula algebra. If the definition of an LT-algebra is scrutinized something very interesting pops out. For the forcing LT-algebras take a class of covers K. Then consider the equivalence relation relative to K, K . K D fŒ˛ W ˛ 2 Sg but ˇ 2 Œ˛ ” ˇ K ˛, and that is just to say that for all F 2 K, mngF .˛/ D mngF .ˇ/. Breaking this definition down further we can see that what really matters is that the meanings relative to the models in F are equal. Take K a class from M  , and Km D fM W M 2 Fg for each F 2 K, then ˛ K ˇ ” ˛ Km ˇ. In this way every forcing LT-algebra corresponds to an LT-algebra of X . But does it go in the other way too?

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121

Theorem 5. The Lindebaum-Tarski algebras for forcing and the logic X are the same. Proof. The comments above imply that the forcing LT-algebras are a subclass of the LT-algebras for X. Take a set of models of X say U . For each M 2 U generate the unit cover hMi. Then the collection of these unit covers forms a class of covers. The LT-algebra corresponding to that class of covers is of course a forcing LT-algebra. 

7.5

Denials

A denial of a formula ' is a formula such that CONX .'; / (the pair set is inconsistent). According to this definition of denial, a logic which has only ?, with its usual meaning, as the only denial will be said to have denials. Formulas like ? are called absurd formulas. A logic X has non-trivial denial when there are denials of formulas other than absurdities. A formula is contingent just when it is neither absurd nor a theorem (or logical truth). Suppose that ' is a contingent formula, and there is another contingent formula such that the pair set is inconsistent. These will be called ‘contingent denials’. The next definition is more complex. Definition 16. Let ˛ be a contingent formula of a logic X and ˇ be a denial of ˛. ˇ, is a Negation-Denial of ˛ (ND) if and only if, 1. ˇ is contingent, 2.

CON X .˛; ˇ/

and,

3. if CONX .˛; ı/ then ı `X ˇ.

We have chosen the term ‘ND’ rather than ‘negation’ since it may be the case that a logic has a connective in its language which is commonly called negation; but, there may be a derived connective which satisfies

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the definition of ND.9 A derived connective is a formula schema that does the work of a connective. A formula does the work of another when the inferential relationships of the two are the same. For instance in a classical system with only ^ and : the derived connective :.:A ^ :B/ does the work of _. The formula :.:A ^ :B/ obeys the same truth conditions as A _ B and it will obey the same introduction and elimination rules in some calculi.10 Notice that ND is not a symmetric relationship. For instance the negation of intuitionistic logic (IL) satisfies this formulation, but A is not an ND of :A in IL. That is because although A  ::A the implication does not go in the other direction. In the sequel we make the following assumption about the logic X: Suppose that CONX .€; ˇ/. Then there is a denial ı of ˇ such that € `X ı. If one further assumes that ND is a symmetric relationship then the logic X will have the following property: [Den*] 8€; ˛; ˇŒˇND˛ H) .€ `X ˛ ”

CON X .€; ˇ//

Note that ND is used to represent the relation of negation-denial. Proposition 3. Given that a logic X has the following properties: 1. If CONX .€; ˇ/ then there is ı such that CONX .ı; ˇ/ and € `X ı. 2. X has symmetric ND’s. Then the logic satisfies [Den*]. Proof. Suppose that ND is symmetric for X, and the other assumptions 1. and 2. hold of X . Then assume ˇND˛. Suppose that € `X ˛. Then, of course, CONX .€; ˇ/ since ˇ is a denial of ˛. Now suppose that CONX .€; ˇ/. Then there is some denial of ˇ, , such that € `X by assumption. Since ND is symmetric, ˛NDˇ. By definition of an ND it follows that `X ˛, and by [Cut] € `X ˛.  9 Consistency

preservation can characterize a logic under certain assumptions on denials. For details see Payette and Schotch (2006). 10 See the Fitch style system in Schotch (2004).

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123

7.6 Forcing and Preservationism The theme of this Volume is to investigate inference relations according to what property or properties they preserve. The obvious question that one might ask is what does this new forcing relation preserve? As mentioned elsewhere in this Volume forcing preserves level. To see this note that the level function is monotonic, i.e., if €   then `X .€/  `X ./. This follows from CONX being downward monotonic–I omit the proof. Theorem 6. If `X .€/ D  then `X .CŒ .€// D . Proof. Again the level 1 case is trivial. Suppose `X .€/ D  ¤ 1. Then let C be a cover of € such that w.C/ D `X .€/. Suppose that 2 CŒ .€/. Then, by definition, C  for all C, of appropriate width, which cover €. However, was arbitrary, as was C. Thus, one can see that C is a cover of CŒ .€/, and has width . Thus `X .CX .€// has level at most , but `X is monotonic, and €  CŒ .€/; therefore, `X .CŒ .€// D .  The relation of forcing as a paraconsistent inference relation is ‘nice’ in the sense that one does not lose out on consequences that  may have to give up. If it is the case that a set of premises is X -consistent then the forcing closure is the closure of the set relative to the logic X . Theorem 7. If `X .€/ D ,  2 f0; 1g then CX .€/ D CŒ .€/. Proof. Assume that ˛ 2 CŒ .€/. Then for any partition C there is i 2 C such that i  ˛. Then, i  €, and, by monotonicity of X, it follows that ˛ 2 CX .€/. Now suppose that `X .€/ D 0, then h¿i is a cover of €. In fact it is the only cover, since it is the only cover of width 0. So CŒ .€/ D CX .€/. Now assume that `1 .€/ D 1. One may restrict the investigation of the covers of € to the partition covers by lemma 4. Thus, all one must consider is the cover C D h¿; €i. The result follows.  So for consistent sets the relation collapses into the underlying logic.

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What about other levels beyond the finite into the transfinite? The level function can assign the level ! to a set under the assumptions which have been made. And of course the forcing closure will preserve that level, but can we say anything about what that closure will be? Recall that the language which we are working with is denumerable at most. S Proposition 4. If `X .€/ D ! then CŒ .€/ D 2€ CX .f g/. Proof. Assume that `X .€/ D !. Then the largest the cardinality of € can be is !. Thus, the cover Cm € is a cover of width !, the level of €. m Clearly C€  CŒ .€/. Let ˛ 2 CŒ .€/. Then for every cover C of  width the level of €, C  ˛. Thus, ˛ 2 Cm €. Until now, the only assumptions that have been used are that the consequence relations, and/or operators, obey [M], [R] and [Cut]. No concerns about compactness or connectives other than that the logic X has connectives and is compositional have been addressed. Now it is time to ask these difficult questions. Namely, what structural rules are preserved in the “move up” and what can be said about compact logics, and the effects of connectives on forcing?

7.7

Structural Rules and Compactness

Continuing with the study of the properties inherited from the underlying logic one might ask if the forcing relation has [R], [M] and [Cut]. That forcing has [R] is perfectly obvious, but that it has [Cut] is not. Before [Cut] is discussed we must discuss [M]. There is a problem with [M]. If € ˛ then it is possible that the level will rise if one adds a set to it. For instance, if the underlying logic is classical then fA; Bg will force A ^ B but, fA; B; :Bg will not force A ^ B, since f¿; fBg ; fA; :Bgg is a partition of width 2, the level of fA; B; :Bg. But that cover will not cover A ^ B. This is the downfall of forcing. It is not monotonic, but it is not anti-monotonic. With a caveat it is monotonic. That restriction is that the set being added must not make things worse. That means that when one adds a set of premises, the new set must not be more inconsistent. The level of the superset must not

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125

change, i.e., `X .€/ D `X .€ [ /. The relation  does not suffer from this; it has all three structural rules. For forcing one needs something  11 different.  The new version  of [M], [M ] is: If `X .€/ D `X .€ [ /, and € ˛ then € [  ˛. However, this is not something assumed about forcing; it follows from the definition.  Proposition 5. The forcing relation , has [M ], given that X has [M].  Proof. Assume `X .€/ D `X .€ [ /, and € ˛. Then, let C  € [ , and w.C/ D `X .€ [ /. Then C  € so by definition C  ˛, since w.C/ D `X .€/. Hence € [  ˛, since C was an arbitrary cover of the proper width.  Given [M ], one can see that the version of [Cut] given in the section above is not satisfied in all cases. However, the formulation of [Cut] as: If € ` ı for all ı 2  and  ` ˛, then € ` ˛ is also not satisfied. Also mentioned in the section above, the version of [Cut] just mentioned gives the correlation between the consequence operator and the consequence relation. Consider the following counter example. Let  D fA; Bg and € D fA; B; C; :C; Dg. Then although any cover of € of width 2 is a cover of , it may not cover  properly. The set  forces A ^ B, but € does not. Thus [Cut] cannot be satisfied, at least on the formulation that has been chosen. But not all is lost, just most of it. A different, but not finitist, version can be satisfied. In fact this principle has been used implicitly many times over in the proofs above. It is as follows: If € ` ı for all ı 2  and €;  ` ˛ then € ` ˛. This will be known as [Cut ].   of X has [R], [M] and [Cut] then

Proposition 6. If the relation   has [Cut ]. That is if € ı for all ı 2  and €;  ˛ then € ˛.   Proof. Assume € ı for all ı 2 , and €;  ˛. Then assume that  C  €, and w.C/ D `X .€/.   CŒ .€/ so C   by definition because C  €. Since `X .€/ D `X .CŒ .€//, proposition 6, it must be that `X .€ [ / D `X .€/. Thus C is a cover of € [  of the right width to be a cover of ˛ by definition of forcing. Since C is an arbitrary  cover of € of the right width, € ˛.  11 This

is also known as X-level preserving monotonicity in Payette and d’Entremont (2006).

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It would be at best misleading and at worst cowardly to omit mention of some rather odd results that flow from the definition of forcing. Forcing can take a logic X that does not have [Cut],12 and give it [Cut ]. Because of this peculiarity, for the logic and its derived forcing relation to agree on consistent sets the logic must obey the structural rules. However, the same cannot be said about the rule [R]. The case of [M ]is similar to that of [Cut ]. The restricted version of monotonicity may hold even though X does not have [M]. But we place these problems out of bounds because the logics we are considering are ‘structural’ as in definition 6. The preservation of compactness is not from the logic to forcing, but, rather, from the logic to the level function. The forcing relation is compact given the compactness of X, albeit in an uninteresting way. If  € then take a partition cover of €, C. By definition of forcing there is a i 2 C such that i  . By the compactness of X we get a 0i ! i such that 0i  . Here ! is the finite subset of relation. However, 0i ! € so we satisfy compactness. Level compactness is much more interesting. For level compactness we need a theorem first. These theorems originally appeared in Payette and d’Entremont (2006), and they will also be proved here. We assume that the consequence relation of the logic X is compact. That means that the CONX predicate is also compact. Theorem 8. The following are equivalent when `X .€/ < ! and CONX is compact:(The subscripts will be omitted.) 1. `.€/  n ” 8€ 0  € that are finite, `.€ 0 /  n. 2. If `.€/ D n then 9€ 0  € which is finite, such that `.€ 0 / D n. 3. If there is a finite subset €  of € such that for any other finite € 0  €, `.€ 0 /  `.€  / then `.€  / D `.€/. Proof. We proceed by showing the equivalence in a triangle. All n; m; k etc. are elements of !. 12 Although, strictly speaking, some might wish to argue that there is no such thing as a logic which fails to have [Cut].

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127

1 ) 2 Assume 1 and assume for reductio that `.€/ D n and that there is no finite subset of € which has level n. We know by monotonicity of level that all of the subsets of € must have level less than that of €, so there is an upper bound. This upper bound will also apply to finite sets. Call this upper bound m. This m is strictly less than n because otherwise there would be a finite subset of level n which there isn’t. With 1 it follows that `.€/  m < n, which is a contradiction. 2 ) 3 Assume 2 and the existence of a €  as in the antecedent of 3. There must be, by 2, a finite € 0  € with `.€ 0 / D `.€/ but then, `.€/ D `.€ 0 /  `.€  /. By monotonicity of ` so `.€  /  `.€/ hence, `.€  / D `.€/. 3 ) 1 Assume 3. The only if direction of 1 follows from monotonicity of ` thus, assume that for every finite € 0  € `.€ 0 /  n. Let m D maxfkj`.€ 0 / D k & € 0  € f i ni t eg This must exist since there is an upper bound, viz. n. Let €  D € 0 such that `.€ 0 / D m. Then the conditions for 3 are satisfied thus, `.€  / D `.€/ D m  n.  With this theorem we can prove: Theorem 9 (Finite Level Compactness). If € is a set of formulas with `.€/ < ! and CONX is compact then: If `.€/ D n then for some finite subset € 0 of €, `.€ 0 / D n. Proof. We will proceed by induction on the level of € and use theorem 8. For `.€/ D 0 then € D ¿ or €  CX .¿/, so ¿  € and is finite with `.¿/ D 0, and by the downward monotonicity of CONX any nonempty subset of a nonempty € will have level 0. The result for the basis step follows. Assume: if `.€/ D k then, For some € 0 ! €, `.€ 0 / D k for k  n. Assume that `.€/ D n C 1. We will show that 3 holds to get the result for this stage. If there is a finite set then we are done, so assume that there is no finite set with level n C 1. All finite subsets must have level (strictly) less than n C 1 by monotonicity of level. Since there is an upper bound on the levels of the finite sets, call it m, m  n < n C 1 and let €  be a finite subset with level m. Then by induction hypothesis using 3 above we get `.€  / D `.€/ D m < n C 1 but that is a contradiction. Therefore for all n 2 !, if `.€/ D n then, for some finite subset € 0 of €, `.€ 0 / D n. 

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128

Notice the crucial restriction to finite levels in these two theorems. The level of a set is bounded by its cardinality. If a set has a level that m is not 1 then Cm € is a cover. But, w.C€ / D j€j. If € had a level m greater than its cardinality, C€ could not be a cover. So `X .€/  j€j. If `X .€/ D ! for example then there is no finite subset that could have level ! since the set can have at most a finite level. What this means is that any set with level ! will have a subset of every finite level. To conclude the general look at properties that are preserved in the move to forcing, consider a particular example of a logic that meets these properties: classical first-order logic.

7.8

First-Order Logic and Forcing

The properties studied thus far can be used to judge the ‘goodness’ of this method of paraconsistentizing a logic. However, they are not just idle properties; they have applications that imply more reasons to think that forcing is a good method. We will show this in two tiers. First we will give the general case then a specific case: first-order logic. What we want to show is that forcing under certain conditions will be the largest relation that can be defined that has many of the aforementioned properties. In particular the key properties will be the restricted versions of the structural rules and level preservation. To accomplish this we must introduce some new objects. Definition 17. € C is a maximal level preserving extension (MLPE) of € if and only if: 1. €  € C , 2. `.€ C / D `.€/, and 3. for any formula

, if `.€ C [ f g/ D `.€ C / then

2 € C.

It follows, with the help of theorem 9 above, that any set with finite

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level can be extended to an MLPE . We can show this by considering the following result about partial orders. Definition 18. A partially ordered set hS; i is a set S with a relation  such that for a; b; c 2 S 1. a  a (Reflexivity) 2. If a  a and b  c then a  c (Transitivity) 3. If a  b and b  a then a D b (Anti-symmetry)

A monotone operator on a partial order, hS; i, say, is an operator, ˆ W S ! S such that if a  b then ˆ.a/  ˆ.b/. Given a set T  S one says that T has an upper bound if, there is an a 2 S such that for all t 2 T , t  a. A least upper bound for T is an upper bound a of T , such that for any other upper bound b, a  b. An element m 2 S is maximal just when, for any a 2 S, if m  a then m D a. A fixed point of the operator ˆ is an a 2 S such that ˆ.a/ D a. A ‘chain’ in S is a set C  S such that for each a; b 2 C , either a  b or b  a. A smallest member of S is a b 2 S such that, for any a 2 S, b  a. With this terminology in mind the following can be shown. Theorem 10. Fitting (1986, pp. 78-9) Let hS; i be a partial order such that S has a smallest member, every non-empty set of members of S which has an upper bound has a least upper bound, and ˆ is a monotone operator on S. Suppose every chain in S has an upper bound. Then, if a 2 S is such that a  ˆ.a/ then there is a maximal fixed point of ˆ, call it b, such that a  b. Proof. Suppose that hS; i is a partial order which satisfies the conditions of the theorem. Let ˆ W S ! S be a monotone operator. Suppose that a 2 S and a  ˆ.a/. Define E D fx 2 S W x  ˆ.x/g. Clearly a 2 E. If one restricts  to E there is another partial order. Suppose that C is a chain in E. C will also be a chain in S thus it has a upper bound, and since it is a nonempty

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W set inWS it has a least upper W bound call it C . Thus for all x 2 C , x  C so ˆ.x/  ˆ. C / since ˆ is monotone. Since x W  ˆ.x/ W because C  E thus for all x 2 C x W  ˆ. C /. So ˆ. C / is an upper bound for C in S so because C is a least upper bound W W W C  ˆ. C /. Thus C 2 E. Hence al chains in E have upper bounds in E. Then apply Zorn’s lemma to get a maximal element extending any element of E, including a. Suppose that m is a maximal element of E. Then by definition of E, m  ˆ.m/ and by monotonicity of ˆ, ˆ.m/  ˆ.ˆ.m// so ˆ.m/ 2 E. Since m is maximal in E, i.e. there can be no proper, or non-equal, extensions of m in E, ˆ.m/ D m (m is a fixed point of ˆ). Take any fixed point, say g, of ˆ in S then g D ˆ.g/ so g  ˆ.g/ thus, g 2 E. Since m is maximal in E no fixed point in S can properly extend m so no maximal element of S can extend m because maximal points must be fixed points of ˆ. Therefore m is maximal in S. Which is what is wanted.  In the logics we are considering, all of the suppositions of the theorems are satisfied, save the condition on chains. For a compact logic the chain condition follows immediately. However, this theorem applies very easily to some propositional logics. But for first-order logics it is not so clear; a maximally consistent extension may not correspond to a model in some cases. We may need something more, but that would take us off track. What we want to show is that a partial order can be made out of sets so that the maximum items will be the MLPE ’s. Consider a set  with a finite level relative to the compact logic X . Define the set L as set of all level preserving supersets of . That is the set L D f€ [  W `X .€ [ / D `X ./g Proposition 7. hL ; i is a partial order which satisfies the conditions in theorem 10. Proof. Any non-empty set D of L that has an upper bound € 0 will be S such that `X . [ € 0 / D `X ./, and,Ssince D  € 0 , and level is monotonic D has a least upper bound, D. That unions of chains have upper bounds follows from the Finite S Level Compactness theorem, since, if the union of the chain C , C ,

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S has the property that `XS . C [/ > `X ./, then there is a finite subset of this set that has `X . C [ /, or there S is some finite subset that has a level greater than `X ./ if the level of C [  is infinite. Either way that finite set would be contained in some set in the chain, but that is impossible. The monotone operator is the forcing closure operator. This operator is not monotone in general, but, since it has [M ], it is monotone on L .  We then have the result that we want: Lemma 5. Let € have level m 2 ! then it can be extended to a € C . Proof. For any €  S it is the case that €  CŒ .€/. Apply proposition 7. It is clear that the maximal sets in this are the upper bounds we get from this theorem. Every upper bound must be in the L so they are level preserving supersets, which takes care of the first and second conditions. The third condition holds because of the maximality of these sets in L : if something could be added to the set that maintained the level, the set would not be maximal in L . The result holds.  Notice that if the level of  is ! then the extensions will be everything except those formulas which have level 1. We can then define the relation which will be the largest on a compact logic, even for sets that the following  of infinite levels. Notice holds for MLPE ’s: € C ” 2 € C . This follows at once since forcing obeys [R], and forcing is level preserving. The largest relation, regardless of underlying logic X, can be defined as follows: \ € `MLPE ' , ' 2 €C € C 2MLPE

where MLPE stands for the class of maximal level T preserving extensions of €.13 We abbreviate the right hand side as € C . 13 The general case will have to obey some other condition if the logic is first order namely the one like in the completeness theorem of Henkin. That is one only considers formulas which occur in the original language of € not the extensions.

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Proposition 8. Given a compact logic X and a relation Y which obeys   [Cut T C], [R], [M ] and preserves X -level we have that for all € CY .€/  € . T C Proof. Assume for reductio that ' 2 CY .€/, but ' 62 € . Then C C there is an MLPE of €, call it €' , such that ' 62 €' . By definition `.€'C [ f'g/ > `.€'C / D `.€/. And since €  €'C , by hypothesis on Y , it follows that CY .€/  CY .€'C / by [M ]. Thus ' 2 CY .€'C /, and so `.CY .€'C // > `.€'C /. Therefore, Y does not preserve X-level. Or, Y does not obey [M ]; either option is contrary to hypothesis.  The next question to raise is whether a sufficient and/or T necessary condition for forcing to agree with this ‘largest’ relation € C , can be established. It can indeed be shown that there is a sufficient condition, but to establish that we need to discuss the object language of the logic X in question. The next theorem holds for logics said to be productival or said to have products. The definition of such a logic is as follows. Definition 19. A logic X is productival iff, given any finite set € there is a formula ' such that ' `X for each 2 €, and € `X '. This notion of a product is like that of a denial. A product will be some derived connective which will allow one to satisfy the conditions above. It is worth noticing that if a logic has binary products, then it has all finite products. This is a well known theorem from category theory. Somebody might be tempted to say ‘Oh I see, products are just a kind of conjunction!’ but that would be to put the matter backwards. In particular, a logic may have products without its underlying language having any explicit connectives at all. Or if it has connectives it might not have an explicit conjunction. Classical logic with _ and : as the only primitive connectives serves as an example. The product of ˛ and ˇ is :.:˛ _ :ˇ/. The further assumption needed about logics is that they have symmetric negation denials. The reader should recall the definition 16. Then we have,

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Theorem 11 (Maximal Forcibility Theorem). Let X be a compact, productival logic, which has symmetric negation denials, so obeys [Den*], and`X .€/ 2 !. Then, € if and only if, for each MLPE € C of €, €C . Proof. ())  (I will omit the subscript X’s.) Suppose € . Then let € C be an MLPE of €. So, `.€/ D `.€ C / by definition. Suppose that C covers € C and has width `.€/. Then C is also a cover of € because €  € C . But, the width of C is also the level of € so, C is a cover of € of the appropriate width for there to be a i 2 C such by the  that i  definition of forcing. Since C was arbitrary € C . (() By contrapositive. Assume €Œ6 and `.€/ D m. Then, there is an C which is a partition cover of € of width m where 8 i 2 C, i 6 . Thus, by [Den*] CON .i [ f 0 g/ for each i, 0 a ND of . For each i ¤ j , i \ j D ¿. And CON .i [ j /. By compactness of CON , for each pair of cells there are finite sets 0i and j0 , contained in i ; j respectively, which are inconsistent with one another. Thus, form their respective products and get 'ij and 'j i . So it follows that CON .f'ij g [ f'j i g/. There are only finitely many of these 'ij ’s for each i; thus, there is a product for each i, call it 'i , such that 'i  'ij for each j ¤ i. The 'i ’s are clearly consistent, by definition of products, with the i ’s, and any two distinct 'i ’s are inconsistent. Use the fact that CON .i [ f 0 g/ to form the product of f'i ; 0 g to get 'i for each i, which will also be consistent with each i . Form the cover: C0 D h¿; i [ f'i g W i 2 C; & 1  i  mi ˚ C0 is a cover of € [ f 0 g of width m. Let €  D € [ 'i W 1  i  m By monotonicity of level it follows that `.€  / D `.€ [ f'i W 1  i  mg/ D `.€/. This new set €  will have a MLPE € C . € C will have level m by definition, and, since each 'i entails 0 , and must be contained in a different cell, cannot be added to € C without increasing its level. But that is to say: € C Œ6 with €  € C .  What this theorem implies is that for logics satisfying the specified properties what is forced by a given set is all that ends up in every MLPE

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T T of the set. Thus forcing agrees with € C , and € C is the biggest relation there is that fits our description. The reason for the assumptions of symmetric ND’s and products is to guarantee that the extensions will be able to exclude a given formula— the one which is not to follow from the set. If given the set fA; :Ag, this set does not force :B, but without the ability to make the formulas, :A ^ B and A ^ B, one may be able to add :B to the set without raising the level. The symmetric ND’s are needed so that entailment and consistency will commute in the right way, i.e., so that non-provability will imply that the set is consistent with an ND. This will of course fail for intuitionistic logic. The general problem is that, depending on the logic, the maximal extension may include things that do not follow from the sets. In IL there are formulas that are consistent with a set, make it into every maximal extension, but are never proved by the original set. These restraints, if absolutely necessary, greatly limit the number of logics for which forcing is the largest relation, but it does not push out all but classical logic. The Łukasiewicz many-valued Łn logics all have conjunctions, and have a type of bar negation which act a symmetric ND. Next comes a particular example of these properties. Like the maximal Henkin-extensions of first-order logic used to prove completeness we need a more complex extension for forcing. C Definition 20. An  -MLPE € ofCaset € is such that: it is an C MLPE and if € .9x/ then € .a/. That is every existential claim is witnessed.

Theorem 12. If  is the relation  of CFL, then for all €; '  S, with C ' for all - MLPE ’s € C such that finite level, €

' ” € €  € C. To prove this we need another extension lemma, one that says that each € can be extended to an -MLPE . Here the reader should also recall definition 7. Lemma 6. Each € with CFL-level n 2 ! can be extended to an MLPE .

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135

Proof. Let `CFL .€/ D n 2 !. Let 1 ; :::; k ; ::: such that k 2 ! be an enumeration of the formulas of the language. Form sets †n for n 2 ! by: †0 D € 8 †n 1 [ f n g if `.†n 1 [ f n g/ D `.€/ ˆ ˆ < †n 1 [ f n g [ f'.a/g if `.†n 1 [ f n g/ D `.€/ †n D and n D .9x/' ˆ ˆ : †n 1 otherwise Where a is notS mentioned in †n 1 [ f n g Let € C D n2! †n . Claim: this set is an -level preserving maximal extension of €. By the recursive construction and compactness `.€ C / D `.€/. First it must be established that for each clause of the construction `.†n / D `.†n 1 /. For the first clause it is simple, by monotonicity of level. Similarly for the third clause. The second clause is more complicated: Assume the result holds. There are three cases if `.€ C / ¤ `.€/. 1. `.€ C / D 1 2. `.€ C /  ! 3. ! > `.€ C / > `.€/ If 1. then an absurd formula was added which cannot be covered, but that is impossible since that would have occurred at some stage k and would raise the level from the previous stage. If 2. then for any n 2 !, there is a finite subset of € C of level n. If there was a finite upper bound on the levels of the finite subsets of € C then the compactness theorem, theorem 9, says the whole set would have that finite level, which is contrary to assumption. So choose n D `.€/ C 1, the finite subset of level n will be contained in some †k so `.†k / > `.€/ which is impossible. Finally, if 3. there was a finite set which was the culprit, and it would be contained in some †n , which is also impossible. Thus `.€ C / D `.€/. Claim: € C is an MLPE . Suppose `.€ C [ f'g/ D `.€ C /, then either a) ' 2 € or, if not, b) ' D n for some n 2 !. If a) then ' 2 € C

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a fortiori. If b) then ' was considered for membership at stage n, and since adding it to € C does not change its level, adding ' to †n 1 does not change the level of †n 1 since `.†n / D `.€/ D `.€ C /. Thus, ' was added at stage n. Therefore, ' 2 € C .  Claim: € C is -complete. Suppose that € C .9x/'. Then `.€ C [ f.9x/'g/ D `.€ C / so, by the previous result, .9x/' 2 € C . And for some k 2 ! .9x/' D k , so k was added at stage k, but then so was '.a/ for some constant a, by the construction. Therefore  € C '.a/ a fortiori. This completes the proof. Lemma 7. The second clause of the construction preserves level. Proof. This uses theorem 9 essentially. Let †n D †n 1 [f n g[f'.a/g with n D .9x/' as is stipulated. By assumption `.†n 1 [ f n g/ D `.†n 1 / D k 2 !. For any †0 ! †n 1 [ f.9x/'g, `.†0 /  k by monotonicity of `. Note that for any †0 ! †n 1 [ f.9x/'g the level of †0 [ f.9x/'g must also be  k. Claim: For all †0 ! †n 1 [ f.9x/'g, `.†0 [ f'.a/g/  k. Suppose not, then there is some †0  †n 1 [ f.9x/'g, such that for each cover C D h¿; 1 ; : : : k i with w.C/ D k, we have CON CFL .i ; '.a// for each 1  i  k. If there wasn’t, then by compactness the result would follow since each finite subset would have level  k. However, CFL obeys [Den*]; thus, one has i  :'.a/ for 1  i  k. Note that a is not mentioned in any i , since C is an elementary cover, and a is not mentioned in †n 1 [ f.9x/'g by hypothesis, so i  .8x/:'.x/ by the rules for classical logic. But that means, i  :.9x/'.x/ also by classical logic. But, some i must entail .9x/' because C is a cover of †0 [f.9x/'g. So, for some i , CONCFL .i /, which is contrary to the hypothesis that C is a cover. Therefore, each finite subset of †n 1 [ f.9x/'g [ f'.a/g has level  k, and by compactness, i.e., theorem 9, the whole set must have level  k. However, `.†n 1 / D k, and †n 1  †n 1 [ f.9x/'g [ f'.a/g so by monotonicity of level, `.†n / D `.†n 1 /. Which is what is wanted.  Using these results the proof of theorem 12 follows easily.

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137

Proof  of theorem 12. Assume that € has finite level. Then suppose that € . Then every -MLPE will force it, since it must contain . Suppose that €Œ6 , then proceed as in theorem 11 by conjoining the proper finite sets with the negation of . Then by lemma 6 extend this new set to an -MLPE which will not force since it would raise its level to do so.  This concludes the study of forcing as an approach to paraconsistent logic that derives a paraconsistent relation from an inconsistency intolerant logic X.14

7.9

Conclusion

This chapter’s purpose is to display many properties that can be preserved by the move to the forcing version of some underlying logic (which is assumed to have the properties). Even when the full version of the property fails (which is guaranteed since the whole point of a forcing relation is to fix perceived problems with the underlying relation) there are usually restricted versions of the properties which hold for forcing. In fact the restricted versions can be seen to make more sense for a paraconsistent relation, which is certainly the case for monotonicity and substitutionality (in the strong sense). The hope is that these considerations may persuade the reader that this method is, from a technical standpoint, the right way to go. What is important to recognize is that the method does not completely change the rules of the game. It recognizes a failing, and then offers a solution. It does not change the meaning of logical constants; it merely does not permit one to use those constants in certain contexts: namely inconsistent contexts. In all of the consistent contexts one may proceed as one always does using the logic one wants. What happens in the inconsistent contexts is that something has gone awry. But, we want to keep as much around as possible of the original way of doing things, for which we presumably have strong motivation. 14 Please note that Theorem 8, Theorem 9, Proposition 8 and Theorem 11 were originally published in Payette and d’Entremont (2006).

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Forcing allows us to keep within sight of the familiar even while straying into unexplored logical territory.

Eight

Representation of Forcing D ORIAN N ICHOLSON

AND

B RYSON B ROWN

Abstract This essay shows how the forcing relation with underlying logic X might be represented in a way which more closely resembles an axiomatic approach. Following the initial result we take up the case of representing the forcing relation in which the underlying logic X allows sets on the the right of `X . This requires us to redefine the notion of X -level forcing to take into account the ‘handedness’ of sets. We must also expand the definition of X-level to take this difference into account.

8.1

Introduction

Once it is realized that X-level forcing1 inherits only some of the principles of the underlying logic X and in particular that an abridgement of the structural rule [Mon] of monotonicity is required,2 the question of alternative presentations of the relation naturally arises. The goal is to give something like an axiomatization of the relation, and once that has 1 See

‘On Preserving’ in this volume for the definitive account. fact many, otherwise well-intentioned folk, have gone so far as to equate monotonicity with deduction, in the sense that if a relation is not monotonic, then it can’t be deductive inference. 2 In

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140

been accomplished, we can, those of us who were worried, heave a sigh of relief. The project is not so urgent as the corresponding problem in so-called n-ary modal logic, but that shouldn’t be regarded as an excuse for not undertaking it.

8.2

Background

We consider logics X which satisfy the usual structural rules: [R] ˛ 2 € H) € `X ˛ [Cut] €; ˛ `X ˇ & € `X ˛ H) € `X ˇ [Mon] € `X ˛ H) € [  `X ˛ For these logics, we understand the notion of consistency in the manner of Post. € is consistent, in or relative to a logic X (alternatively, € is X -consistent) if and only if there is at least one formula ˛ such that € °X ˛. Where X is a logic, the associated consistency predicate (of sets of formulas) for X , is indicated by CONX . Having a notion of consistency allows us to define a certain relation between pairs of formulas (of the underlying language of the logic X — after this we won’t bother to keep mumbling that particular mantra) which we term non-trivial denial. X has non-trivial denial if and only if for every non-absurd formula ˛ there is at least one non-absurd formula ˇ which is not X -consistent with ˛. Two formulas which are related in this way are said to deny each other. We also say each is a denial of the other. There are, in addition, two properties which we may require of a logic X :

8.2. BACKGROUND

141

We shall say of the logic X that it is productival (or that it has products), which is to say two things: Q Q ( 1 ) For every finite set † there is a formula .†/ such that Q .†/ `X i for every i 2 † and Q Q ( 2 ) † `X .†/ We say of the logic X that it is coproductival (or that it has coproducts) when it satisfies these two conditions ` ` ( 1 ) For every finite set of formulas † there is a formula .†/ such that ` for every  2 †,  `X .†/ ` ( 2 ) If, ` for any formula ˇ,  2 † H)  `X ˇ then .†/ `x ˇ When the logic X enjoys all of these, it will be said to have a full set of properties. Given a set €, of formulas drawn from the language of the logic X, we say that an indexed family of sets (starting with ¿), is an X-logical cover of € provided: For every element a of the indexed family, and S € CX .ai /

CON X .a/

i2I

where CX .ai / is the (X ) logical closure of ai which is to say fˇjai `X ˇg. When F is an (X ) logical cover for € we write COVX .F; €/. Mention of the particular logic X is often suppressed when no confusion will be thereby engendered. We shall refer to the size of the index set of the indexed family in this definition as the width of the logical cover, indicated by w.F/. And now for the central idea: The level (relative to the logic X) of the set € of formulas, indicated by `X .€/ is defined:

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`X .€/ D

8 < min ŒCOVX .F; €/ if this limit exists w.F/

:1 otherwise

In other words: the X-level (of incoherence or inconsistency) of a set † in a logic X is the width of the narrowest X-logical cover of †, if there is such a thing, and if there isn’t, the level is set to the symbol 1. Finally we define the relation of X-level forcing which, as its name suggests, is derived from some underlying logic X. The relation is defined:  € X ˛ if and only if, for every division of € into `X .€/ cells, for at least one of the cells ;  `X ˛

8.3 Representation  In order to represent, as we shall say, the relation X of X -level forcing, we shall provide another relation to be called simply X-forcing, indicated by `X and eventually show that the two are the same relation. In doing this we axiomatize, in a certain sense, the relation of X -level forcing for a variety of logics X. X -forcing  We describe `X by saying first that it connects to X by means of the singular sets of formulas in this manner: [Sing] If € is singular then  € `X ˛ ” € `X ˛ where a set is singular if it cannot be broken up by logical covers. In this case the empty set is singular, along with all unit sets. As a result of this bridge rule, we should notice the consequence:    [] f g `X f˛g & f g `X fˇg H) f g `X  f˛:ˇg

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143

Next we stipulate that X-forcing satisfies [Ref] and [Cut] but only a restricted form of monotonicity (or dilution) which has two parts:   [M1] If € `X ˛ then € [ €  `X ˛ provided `X .€/ D `X .€ [ €  /  [M2] If € `X ˛ then € [ €  `X ˛ if € is singular. We shall have need of bunches of products as well as coproducts of such bunches. To maximize the ease of displaying such conglomerations of formulas we define: ˚ Definition 1. €ij for € n i ; j

Definition 2.

PAIRS .€/

for the set of pairs of elements of €’

For example, if our set is fx; y; zg, then PAIRS.fx; y; zg/ D ffx; yg ; fy; zg ; fx; zgg Definition 3. PPAIRS.€/ for the set of products of pairs of elements of € For example PPAIRS.fx; y; zg/ D f… fx; yg ; … fy; zg ; … fx; zgg We  the aid of this notation, we can now state two rules of inference for `X .   [RPig] €  `X ˛ & : : : & € `X ˛k H) 1 ` € `X fPPAIRS.f˛1 ; : : : ˛k g/g ˚  ˚ [LPig] €ij` ; … i ; j `X ˛ for every i ; j 2 PAIRS.€/ H) €; PPAIRS.€/ `X ˛ Where ‘Pig’ suggests ‘Pigeonhole.’

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The assumption of the two rules is that the cardinality of € exceeds the X -level of €. The soundness of [RPig] is no serious issue since it is simply a way of stating the pigeonhole principle. The soundness of [LPig] requires a bit more than just seeing however. The requirement is provided by the following: ˚ Lemma 1. Let €ij D €ij [ …. i ; j / . Then `.€ij / D `.€/, when

i ; j 2 € and there is some X -cover of € which is a cover of their product. Proof. Suppose that ˚ F is an X -cover of € and w.F/ D `.€/ and F is a cover of … i ; j for some i ; j 2 €. Suppose further that `.€/ D k. Now let F D f0 ; : : : k g, and for ˚some ˚ i 2 F, i `X  D  [ … ; . i ; ˚j / as in the hypothesis. Let  , and form i i j i F D 0 ; : : : ; i ; : : : ; k . This is clearly an X-cover of €ij ; thus, `.€ij /  `.€/. Suppose for reductio that `.€ij / < `.€/. By definition there is F0 that covers €ij with width w.F0 / D `.€ij / < `.€/. Let F0 D f0 ; : : : ; m g, and m < k. We want to show that F0 is a cover of €. There is some i 2 F0 that proves . i ; j / since it is a cover of €ij . By definition of product, the product must prove both i and j . And there must be a cell that proves the product so by [Cut] on the underlying logic˚we have that that cell,  i , must prove both i and j . But € €ij D i ; j . So that makes F0 a cover of €. However, that means `.€/  w.F0 / < `.€/, and that is a contradiction. Therefore `.€ij / D `.€/.   Lemma  holds for X ˚ Q ˚ 2. [LPig] €ij`;

i ; j  X ˛ for every i ; j 2 € PPAIRS.€/ X ˛

PAIRS .€/

H)

Proof. Arguing indirectly, we show is that there is some €ij ; . i ; j / that does not X -level force ˛, given that the consequent of [LPig] fails. We do˚ this by showing that there is a cover, of appropriate width, of €ij [ . i ; j / that does not cover ˛.

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145

By assumption, the antecedent of the conclusion does not X-level force ˛, so there must be a cover of €, of width `.€/, that does not cover ˛. Since the cardinality of € exceeds the X -level of €, there must be a cell in this cover˚ with ˚ at least two elements by pigeonhole reasoning. Take the €ij [  i ; j relative to those two elements and that set will not X -level-force ˛. What needs to be shown to give us that conclusion is that the level of this set is the same as the level of €. That way the cover that we construct has ˚ the right width to act as an X -level forcing cover for €ij [ . i ; j / . This follows at once from the previous lemma.  The Representation Theorem With all this under our belt, we are finally ready to prove the Representation Theorem for X-level forcing. We assume that X enjoys a full set of properties. Theorem 1 (Representation Theorem for X-level Forcing). For every set € of formulas and every formula ˛   € X ˛ ” € `X ˛ Proof. The proof is by induction on the cardinality of the premise set €. The basis of the induction is the case in which the cardinality in question is less than or equal to the X -level of the premise set. In this case, since there will be at least one logical cover of the premise set in which each cell contains no more than one formula, there must be an X -inference of the form f g X ˛ and thus the result follows at once by [Sing]. In the induction step, the hypothesis of induction is that the cardinality of the premise set is n+1, and that the result holds for all premise sets of cardinality up to and including n. To avoid loss of generality we must take n to be greater than the X-level of €. ˚  It is easy to see that €ij ; … i ; j X ˛ Assume € ˚ X ˛. for every pair i ; j in PAIRS.€/. Since the cardinality of each of the premise sets is less than of €, by the hypothesis of  the cardinality  induction we may replace X by `X in each of the inferences. But

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 ` by now`invoking the rule [LPig] we get €; fPPAIRS.€/g `X ˛. But € `X fPPAIRS.€/g by [RPig] so the result follows at once by [Cut]. 

8.4 More Definitions and motivation It would be an exaggeration, although a pardonable one, to say that the most general way to do proof theory is to take an inference relation to be a relation between sets of formulas.3 This is different from studying the relation between a set of premises and its closure, which is simply gathering together all of the single formulas which follow from the the premises. In this new approach we, in effect, draw a distinction between two kinds of sets (of formulas). We could label the two kinds, premise sets, and conclusion sets, though that terminology invites a confusion between the notion of deductive closure which is after all a set of conclusions, and the new species which we now consider. For the most part, we call the sets that appear on the left side of the provability symbol, l-sets while those on the right are r-sets. Perhaps the easiest way to think of what is going on here is as a kind of dualism, that r-sets and l-sets are dual to one another in a sense that we shall try to illuminate as we go. As a starting point, we can think of sets on the left as candidates for assertion, and sets on the right as candidates for denial. The consequence relation holds when assertion of all members of the left set is incompatible (according to some logical standard or measure) with denial of all members of the right set. First we shall formulate our basic structural rules, generalizations of the rules which we have already seen, which apply to all the logics we discuss, unless we issue an explicit disclaimer. [R]  \ € ¤ ¿ H) € `  [Cut] €; ˛ `  & € ` ˛;  H) € `  3 Those who favor Gentzen-style proof theory, also known as sequent calculus, would argue that sequents are more general than sets since the latter, in effect, build in a number of structural rules rather than making them explicit.

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147

[Mon] € `  H) € [ † `  [ ƒ In previous work we took the notion of consistency as central. Alas, this is peculiar to l-sets, which is to say that it doesn’t translate to r-sets. Of course, this is what we ought to expect. What we shall require is a more general notion, which we call triviality which comprehends both sides of the provability relation. Here it is for l-sets: A set € is said to be (X) l-trivial, indicated by if and only if € `X ¿.

TRIV XL .€/

Notice that, since we are assuming that `X satisfies [Mon], this amounts to saying that € proves every set, which is simply a way of saying that there is no distinction between what € proves in X and what it doesn’t. Whatever notion of X-provability has interested us thus far can interest us no longer, when it comes to €. For r-sets, we must take the dual formulation. A set  is said to be (X ) r-trivial, indicated by TRIVXR ./ if and only if ¿ `X . l-triviality is of course merely inconsistency in sheep’s clothing while r-triviality seems much more radical or even gratuitous. An r-trivial unit set, say f˛g, which we would normally write ˛, is what others would call a theorem of the logic in question. So far from being trivial, these are the whole point of a logic, aren’t they? In a word, no. Logic is about inference, and the r-trivial sets trample the distinction, dual to the one offended by inconsistency, between what proves some set and what doesn’t. Once again, given [Mon], every set proves an r-trivial one. In general terms we might say this in defence of our derogatory terminology: In so far as an inference relation draws a distinction between those kinds of reasoning which are correct and those which aren’t, both kinds of triviality are well-named. In both cases there is no longer a distinction. In the l-case we have lost the distinction between what a set proves and what it fails to prove and in the r-case the dual distinction between what it would take to prove a set and what it would take not to

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prove it. There is no reason to reject one sort of trampling while embracing its dual. And while we are at it, let us not forget that ‘tautology’ is indeed a term of derogation, often prefixed by some such qualifier as ‘mere.’ First, we define an ambidextrous predicate: Definition 4. Relative to a logic X , a set † is said to be X-consistent indicated by CONX .†/ if and only if TRIVR .†/ if † is on the right, and TRIVL .†/ if † is on the left. Here the overline indicates predicate negation. Having increased the scope of the consistency predicate to cover both kinds of sets, we carry over our earlier definitions of logical cover and level which have become similarly ambidextrous. Level applies to both flavors of set, but rather than use explicit notation to display which kind of level is at issue, we shall let the context determine whether we are dealing with left-level or right. It ought to go without saying that one and the same set will not in general have the same level on both sides of the inference relation.4 Having level functions for both left and right handed sets, permits us to define two derived inference relations. The first is just X-level forcing, while the second, more exotic one we shall dub inverse X-level forcing. Once the second relation is defined, the aptness of the name we have chosen will become apparent. Note that while we speak of divisions rather than covers of € and  here, the effect is the same. X -level forcing:  € X ˛ if and only if, for every division of € into `X .€/ cells, for at least one of the cells a; a `X ˛ inverse X-level forcing:  ˛ X  if and only if, for every division of  into `X ./ cells, for at least one of the cells a; ˛ `X a 4 A significant exception is ¿ which has the same level on the left as it does on the right, namely 0.

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149

Although the inverse case looks a bit unusual, both these relations preserve level, though we need to specify what that means in the inverse case. Here we must define, for every conclusion-set †, the inverse Xtheory of †, indicated by CX1 .†/ as Definition 5. CX1 .†/ D fj `X †g To say that X -level forcing preserves level is to say that `X .€/ D `X .CX .€// and to say that inverse X-level forcing preserves level is to say the same thing except for inverse closure, i.e. `X ./ D `X .CX1 .// And this in turn means that neither left or right-handed sets can be trivial unless the set in question contains an absurd formula. Just as we early showed how to represent X -level forcing, we can now do the same for the inverse relation. Inverse X-Forcing It is relatively easy to see that what we could do by way of representing the relation of X -level forcing, we can also do with the relation of inverse X -level forcing, by ‘dualizing.’ This certainly applies to the syntactical presentation of the relation we call inverse X-forcing. We say, as before, that it connects to X by means of the singular sets of formulas in this manner: [Sing] If € is singular then  ˛ `X € ” ˛ `X € where a set is singular if it cannot be broken up by logical covers. Once again the empty set is singular, along with all unit sets.

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Next we stipulate that inverse X -forcing satisfies [Ref] and [Cut] but these rules should be stated:  [Ref] ˛ 2 € H) ˛ `X €    [Cut] .˛ `X €; ˇ & ˇ `X €/ H) ˛ `X € While the restricted form of monotonicity (or dilution) still has two parts, namely the duals of the previous ones.   [M1] If ˛ `X € then ˛ `X € [ €  provided `X .€/ D `X .€ [ €  /  [M2] If˛ `X € then ˛ `X € [ €  if € is singular. Using the notation we introduced for X-forcing, we can state the dual pigeonhole principles which hold for the inverse relation.   [LPig] ˛1 `X € & : : : & ˛k `X € H) … fCOPPAIRS.f˛1 ; : : : ˛k g/g `X €  ˚ `˚

i ; j for every i ; j 2 [RPig] ˛  `X €ij ; ˛ `X €; … fCOPPAIRS.€/g

PAIRS .€/

H)

The assumption of the two rules is that the cardinality of € exceeds the X -level of €. We use the expression COPPAIRS.†/, by analogy with PPAIRS, to represent the set of coproducts of pairs of elements of †. We can argue for the soundness of the Pig rules and for the representation of inverse X-level forcing by inverse X-forcing, by simply reusing the original arguments for the representation of X -level forcing, in dual form. We now consider the proper way to characterize the set-set version of forcing.

 8.5. € `X 

8.5

151

 € `X 

 We start by describing the relation `X , called X-forcing, based upon the underlying (multiple conclusion) logic X. Our aim is to show that this relation is the same as the multiple conclusion X-level forcing that we describe below. It will turn out that there are two ways to accomplish this, as we have been calling it, representation result, and that the two theorems are equivalent. We shall stipulate that X satisfies the structural rules already mentioned and is productival. We shall also, as we expect from the earlier account, want X to be coproductival which we may now define more elegantly than we did earlier. ` ` ( 1) For every finite set , there is a formula  such that ` For every ı 2 , ı `X  and ` ` ( 2)  `X   We describe `X by saying first that it connects to X by means of the singular sets of formulas as we have done earlier. [Sing] If € and  are singular then  € `X  ” € `X  The notion of singularity need suffer no change from the earlier concept so that the empty set is singular, along with all unit sets. As a result of this bridge rule, we should notice the consequence:    Q [ ] f g `X f˛g & f g `X fˇg H) f g `X  f˛; ˇg Next we stipulate that X-forcing satisfies the multiple conclusion form of [Ref] and [Cut] but again, only a restricted form of monotonicity (or dilution) which has two parts:   [M1] If € `X  then € [ €  `X  [  provided `X .€/ D `X .€ [ €  / and `X ./ D `X . [  /

CHAPTER 8. REPRESENTATION OF FORCING

152

 [M2] If € `X  then € [ €  `X  if € is singular and € `X  [  if  is singular. Finally, we must expand our collection of pigeonhole principles.   [RPigL] €  `X ˛ ;  & : : : & € `X ˛k ;  H) 1 ` € `X PPAIRS .f˛1 ; : : : ˛k g/;   ˚ Q˚ [LPigL] €ij` ;

i ; j `X ˛;  for every i ; j 2 PAIRS.€/ H) €; PPAIRS.€/ `X ˛;   ˚ `˚ [RPIGR] €  `X ijQ ; ıi ; ıj for every ıi ; ıj 2 PAIRS./ H) € `X ; COPPAIRS./   [LPIGR] €; Q ı1 `X  or : : : or €; ık `X  H) €; COPPAIRS.fı1 ; : : : ; ık g/ `X  In the statement of these principles, we assume that the cardinality of the sets € and  are strictly greater than their corresponding levels.

8.6

 € X 

 We shall say that € X  provided that for every pair F€ , F of logical covers of the premise set and the conclusion set respectively, that there is a kind of ‘onto’ relation between them of this form: .9a 2 F€ & exi st sb 2 F W a `X b/ In other words, some cell in each logical cover of the premise set proves (in X ) some cell in each logical cover of the conclusion set.

8.7

The Representation of Forcing

We shall consider two theorems which characterize, or represent the relation `X . Theorem 2. (First Representation Theorem) For every pair €,  of sets offormulas  € `X  ” € X 

8.7. THE REPRESENTATION OF FORCING

153

Theorem 3. (Second Representation Theorem) For every pair €,  of sets  of formulas   € `X  ” for some formula ˛, € `X ˛ & ˛ `X  Lemma 3. (Equivalence of the two representations) For every pair €, of sets of formulas   € X  ” for some formula ˛; € `X ˛ & ˛ `X  Proof. The right to left direction of this equivalence is a simple (in effect) soundness proof which we can omit with a clear conscience. The hard direction, from left to right, we prove by double induction on the cardinalities of € and . In the basis case, the cardinalities of the two sets are less than or equal, respectively to the X-level of € and . In such a case the only  way that € X  can hold is if there is an X proof `X ı between some pair of formulas, say 2 € and ı 2 . By singularity, it must follow that € X ˛ and that ˛ X , for some formula ˛. As the hypothesis of induction, assume that the result holds for cardinalities of €,  up to n. In the first part of the induction step, we assume that j€j is greater X than  ` .€/ (else the result would hold trivially by singularity), and that € X . Now it follows that there must be pairs i , j of ˚ distinct members of € such that the X -level of the set € ij D €ij [  i ; j is equal to the X-level of € and that € ij X . As before ˇ we ˇ indicate by €ij the set € with the pair i ; j removed. Moreover ˇ€ ij ˇ D n 1, so that the hypothesis of induction of some formula,  gives us the existence  let us call it ˛ ij such that € ij `X ˛ ij and ˛ ij `X  If, for each one of these pairs i ; j , we gather together the corresponding results gained from the hypothesis of induction, we shall have, after using a [PIG] rule and the fact that X has coproducts: €;

`

 ` ˚ ij ˛ .i; j / 2 fPPAIRS.€/g `X

PAIRS .€/

 A use ˚ the other [PIG] together with [Cut] results in `of € `X ˛ ij .i; j / 2 PAIRS.€/ But since each element of the coproduct X -proves , so does the coproduct which serves as the formula X -forced by € which X-forces . This ends the first induction step.

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The second induction step is, once again, the dual of the first in which we replace pairs ıi ; ıj from delta with their coproducts, which we must be able to do for at least some pairs, with increasing the level of . The we simply use the same kinds of moves (except dualized) to get the existence of some formula X-forced by € which X-forces .  But this serves as a guideline for the proof of the first representation theorem as follows:  Suppose € X . Then as we showed in the equivalence lemma,   € `X ˛ and ˛ `X .  But then by the allowable uses of monotonicity, € `X ˛;  and €; ˛ `X   But then, by [Cut] € `X .

Nine

Forcing and Practical Inference P ETER S CHOTCH

Abstract In this essay we consider two variations on the theme of forcing. In one of these we restrict the number of possible logical covers of a given premise set by specifying certain subsets, which we call clumps and requiring that any cell which contains a member of a clump, must also contain the other members. The other variation allows for redefining the notion of logical cover in order to require that the cells not only be logically consistent but also practically consistent. In other words in this mode we require of any logical cover that it recognize our theoretical (and perhaps also practical) commitments, by defining the cells to be those subsets (if any) which do not prove (in the underlying logic) any denial of those commitments.

9.1

Introduction

What we might call the pure or perhaps general theory of the forcing relation is all very well in a theoretical setting. That shouldn’t come as much of a surprise since our treatment of forcing tends to be abstract and general. On the other hand, sometimes we want to study inference in something more closely resembling ordinary life, more ordinary than mathematical life at least. In these cases the general theory is too spare. 155

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156

In fact we shall distinguish two ways in which one might wish to come down from the Olympian heights of generality to no greater altitude than a foothill might boast. First a sketch of the general theory:1 We consider logics X which satisfy the usual structural rules: [R] ˛ 2 € H) € `X ˛ [Cut] €; ˛ `X ˇ & € `X ˛ H) € `X ˇ [Mon] € `X ˛ H) € [  `X ˛ For these logics, we understand the notion of consistency in the manner of Post: € is consistent, in or relative to a logic X (alternatively, € is X -consistent) if and only if there is at least one formula ˛ such that € °X ˛. Where X is a logic, the associated consistency predicate (of sets of formulas) for X , is indicated by CONX . Any formula the unit set of which is not consistent, is said to be an absurd formula. Having a notion of consistency, we go on to define a certain relation between pairs of formulas of the underlying language of the logic X (mention of which is often suppressed when it is unlikely to lead to confusion) which we term non-trivial denial. X has non-trivial denial if and only if for every non-absurd formula ˛ there is at least one non-absurd formula ˇ which is not X -consistent with ˛. Two formulas which are related in this way are said to deny each other. We also say each is a denial of the other. 1 The

definitive account is presented in the essay ‘Preserve What?’ in this Volume.

9.2. DIRTY HANDS

157

Given a set of formulas €, we say that an indexed family of sets (starting with ¿) is a logical cover of € provided: For every element a of the indexed family, and S † CX .ai /

CON X .a/

i2I

where CX .ai / is the (X ) logical closure of ai which is to say fˇjai `X ˇg. When F is an (X) logical cover for € we write COVX .F; €/. We shall refer to the size of the index set of the indexed family in this definition as the width of the logical cover, indicated by w.F/. And now for the central idea: The level (relative to the logic X) of the set € of formulas, indicated by `X .€/ is defined: 8 < min ŒCOVX .F; €/ if this limit exists `X .€/ D w.F/ :1 otherwise In other words: the X-level (of incoherence or inconsistency) of a set † in a logic X is the width of the narrowest X-logical cover of †, if there is such a thing, and if there isn’t, the level is set to the symbol 1. Finally we define the relation of X-level forcing which, as its name suggests, is derived from some underlying logic X . The relation is defined:  € X ˛ if and only if, for every division of € into `X .€/ cells, for at least one of the cells ;  `X ˛

9.2

Dirty Hands

Now this is all very well in general, for that most abstract plane of logical existence on which epistemology intrudes not at all upon how we reason. However this is seldom a good representation of real inferential life. When we face the world of applications, there may well be non-logical considerations to weigh in the balance.

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In real life, we have commitments. Some we clasp to our bosoms after striving to acquire them, while others fall upon us unbidden. At this point, some of our opponents may become existentialist enough to declare that they themselves renounce all commitments and refuse to be bound by anything but their own mighty wills. Such as these have put themselves not only beyond polite society, but also beyond science. If the mission statement of science includes the aim of framing bold hypotheses and then suffering these to be tested, science involves commitments. We see this directly we begin to think about how hypothesis testing works. It is clear that, whether we construct a new experiment, or merely use data previously gathered, we have to assume that the laws of science, except of course for one that we may be testing, hold. It simply makes no sense to talk about testing all the laws of science at once. Testing them against what? We are committed, in this endeavor, to the correctness of whatever bits of science are not being tested. This is merely one example, though it is a particularly compelling one. But what does it mean to say that we are committed to this or that principle? This much at least: For the duration of our commitment, we shall brook no denial of the principle in question. In other words, we shall treat any such denial as a sort of practical contradiction.2 Apart from worrying about commitments there are clearly cases when merely having consistent cells is not enough even if we expand our horizons to include practical consistency. Cases that is, in which we want to insist on the integrity of certain of our data. We may be prepared to carve up our premise set into subsets, but we wish to place limits upon how much carving can be done.

9.3

†-Forcing

Whatever else we may mean by commitment, intolerance of any denial of the commitment must be high on the list. There may come a day in which an argument is so convincing that it releases us from one or more of our commitments, but when that happens, it goes without 2 Such was a popular device in Buddhist logic where a paradigm of absurdity was the sentence ‘Lotus in the air’ which we might take as an Eastern version of ‘Pie in the sky.’

9.3. †-FORCING

159

saying we are no longer committed to whatever it was that previously claimed our allegiance. So we aren’t saying that commitment is forever, even commitment worthy of the name. Commitments come and go—an ever shifting pattern as we fail to renew our subscription to one theory and subscribe instead to another. While our commitment is in force however, while we are still bound by it, we shall not suffer any denial to be concluded except in the special circumstances noted below. This is the grand picture at any rate, and like all such, the devil is in the details. One thing to give us pause here is the recognition that our logical commitments are in no way inferior to our other commitments3 which places definite limits upon how much refusing we can do. Think of it this way: We want to treat the denials of commitments, we refer to the set of these by †, as absurd—in some wider sense than logically absurd perhaps, but absurd nonetheless. Since we shall be dealing with these commitments in the manner of forcing, there will be some underlying logic X in terms of which initial inferences take place, Given what we have said so far, no member of † can be a(n X ) consequence of a singular set and still be blocked. Earlier we defined as singular, the sets that suffer no diminution in any logical cover. In other words, our ability to block these bad conclusions has the same limit as our ability in ordinary forcing to block the inference to logical absurdities. But might we not change this? It all depends upon how we choose to regard the notion of singularity for sets. For the most part, we have been thinking of the singular sets as defined as we did just above. If we stick with that line, then the analogy between practical absurdity and logical absurdity continues to hold. On the other hand, we have the option of regarding singularity as a primitive predicate of sets (of formulas) which might be the victim of whatever we want to stipulate in the way of exceptions to the way we have understood the idea previously.4 We might, that is, choose to excommunicate all those sets which 3 Though we wouldn’t follow Quine in giving them a more central place in our ‘web of belief.’ 4 In effect, this was the approach taken when the topic was introduced in Schotch and Jennings (1989).

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were previously singular but consist of unit sets of members of †. To carry out this program of banishment we would also have to limit ourselves to those commitments the denials of which are not X -consequences of the empty set. We shall examine the upshot of this alternative stern approach below, but we should note here the existence of opposition to it. Some of our friends have suggested that striking off some sets previously regarded as singular, is contrary to the spirit of preservationism. The most strongly held intuition behind the notion of forcing is that, in the face of problematic inferences in the underlying logic, we fall back to looking at inference from subsets which are problem free. Under any such regime, those inferences in the underlying logic from sets which cannot be partitioned, must be reckoned correct in the derived inference. To deny this is to break the connection between the derived logic and the underlying logic, a connection which forms a large part of the the justification for preservationism as a revision rather than a replacement theory.5 That is one of the things that separates preservationism from alternative approaches to paraconsistency. We reply that any general scheme for setting aside those X-licensed inferences, even though they must survive partition, would indeed be anathema. But, we are not suggesting anything so general as all that. What we must do is balance our distaste for jettisoning inferences which seem otherwise unassailable, with our distaste for practical absurdities. In the sequel we shall present both approaches side by side. Imagine a case in which there is a physical theory which provides us with plenty of well-confirmed predictions and also, unfortunately, other predictions which are simply absurd, in practical terms at least. Lets say, for instance, that the theory predicts that some physical constant e has an infinite value. What shall we do? A philosopher will immediately say that we have disconfirmed the theory and that we must get rid of it at once. A natural scientist might well have an entirely different slant on things. A scientist is apt to regard a theory as rather like a salami. Late at night while working in the lab, a scientist overtaken by hunger might go to the fridge and take out a salami to make herself a snack. Alas, there is 5 For

a discussion of these terms, consult the first chapter of this Volume.

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161

some mould on one end of the salami. Does she throw it out? Probably not. After all when faced with a salami which has mould on one end, no law compels us to eat the mouldy part. Cut a few slices from the good end! And the same goes for our theory. Don’t use the mouldy part of that either. In fact, why not cut away the mouldy part of the theory entirely and, in order to make it sound better, we could refer to this operation as ‘renormalization.’ Of course it would be better if we didn’t have to resort to this sort of thing, but the world being the kind of place it is, we often do. Show her a physical theory which does all that her renormalized theory does, without invoking that somewhat disreputable operation, and our scientist will embrace it with open arms. Later, at some learned conference or other, she and her colleagues might have a good laugh at the bad old days of renormalization, and proclaim that the younger folk in the profession have it much easier than they did. But right now there is no such theory and the lab is waiting for our experiments and the journals our articles. There are promotions to get and prestigious fellowships. There are even prizes to win and often respectable amounts of cash are involved. Little wonder then that nobody is prepared to twiddle their scientific thumbs while awaiting a more worthy theory. This makes philosophers sound a lot purer in spirit than scientists, but it really isn’t so. Consider the ever fascinating realm of moral philosophy. Here we have a rather similar problem. We have two grand approaches to moral theorizing which we might term axiological and deontological. Each of them ‘sounds right’ in the sense of agreeing with many of our central moral intuitions. But not only are they not compatible, each is refutable by appeal to a central intuition highlighted in the other.6 In other words, we cannot abandon axiology for deontology or vice versa in order to gain a more secure foundation for morals. So it seems in moral philosophy as in physics, we make do. We use the non-mouldy part of the theory and pretend as well as we can that the mouldy part doesn’t really exist. If we generalize our notion of X 6 Thus deontological theories are vulnerable to ‘lifeboat’ counterexamples while axiological theories are vulnerable to ‘scapegoat’ counterexamples.

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consistency, we can make our logic conform to this behavior. We can agree that it is a stop-gap measure, one to be used as a last resort, but evidently a last resort is needed. The central idea is that we get all the mould into one set called †. Less colorfully, † contains the (X) denials of each of our commitments so that every member of † is a sentence which we would like to exclude from the output of our theory. A set € is (X) †-consistent indicated by that € `X ˛ H) ˛ 62 †.7

CON † X,

provided

Evidently, †-consistency implies the usual sort. From this, the definition of †-logical cover and †-level is obvious. In those definitions we merely replace CONX with CON† X . The revised X notions will be denoted by COV† and ` respectively, while the account X †  † of forcing which uses † will be represented by X . It should be clear that there will be, in general, far fewer †-logical covers for a given set € than there are logical covers. Not all of the old rules will hold if we modify singularity by removing members of †; there is bound to be some cost associated with banishing certain contingent sentences. The earlier form of [Ref] had it that: € \¤¿  € X  when sets are allowed on the right, and  ˛ 2 € H) € X ˛ when they aren’t. This will no longer hold when either € \ † ¤ ¿ or  \ † ¤ ¿. What we require is a more general form of [Ref] viz.: 7 In the more general calculus where sets are allowed on the right, this definition can be stated more simply using € °X †.

9.3. †-FORCING

[Ref† ]

163

€ \  ¤ ¿ & .€ [ / \ † D ¿  † € X 

or  † [Ref† ] .˛ 2 € & ˛ 62 †/ H) € X ˛ We get back The usual form of [Ref] by replacing † with ¿. In fact this move collapses †-forcing into vanilla forcing. We also get back [Ref] if we don’t tinker with the description of singularity. Tweaking the notion of singularity also requires other changes. We cannot inherit quite as many inferences from the underlying logic X as we did. It remains true that any X-theorem will also be an X-†-forcing theorem, since the members of † are contingent by stipulation, but it will no longer be the case that every X-consequence of a unit premise set will be preserved. We must drop all those unit sets the members of which belong to †. Of course this means that such sets cannot figure in the generalization to X-forcing of the rule [Mon] of monotonicity. †-forcing is a true generalization of forcing and not one of the those approaches which use something akin to counter-axioms. Neither do we simply add the denials of all members of † to every premise set and proceed from that point by ordinary forcing. The difference is simply that on the counter-axiom approach, or the modified forcing method, the denials of members of † are, one and all, consequences of every premise set. This is not true for †-forcing since, for example, each ‘bad-guy’ in † is contingent, none will be †-forced by the empty set. Neither will any of the denials, presumably ‘goodguys’, although not so good that we want to be able to infer them from an arbitrary premise set. Having said that, we must acknowledge that while †-forcing might provide the basis for some operation like renormalization in physics, it can’t be the whole story. This is because we don’t simply block the unpalatable in the latter case, we also replace it with the palatable. So if the raw theory tells us that the value of e is infinite, we not only snip that out, we replace it with what we take the actual value to be. One can imagine different ways this might happen, including adopting

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extra ‘non-logical’ axioms, but more work would need to be done before something convincing could be offered.

9.4 A-Forcing This is another of the variations on the notion of forcing which reduces the number of logical covers to be considered in calculating whether or not certain conclusions follow from a given premise set. In many applications, unadorned forcing seems to do a bit too much violence in breaking up premise sets (and conclusion sets too, perhaps). If we think of forcing as a candidate for an inference relation to use in reasoning from sets of beliefs, this quickly becomes evident. Representing the belief set of an individual i by Bi , it might be argued that in most cases Bi has a level greater than 1. It might even be argued that Bx must have a level of at least 2 if i is rational. The conflict between this motivation and ordinary forcing or †forcing, is that we usually identify within some Bi natural clumps of beliefs. Thus, while the whole set of i’s beliefs may have level 2 or more, we expect and allow i to draw inferences from certain consistent subsets of Bi . In other words, we refuse to allow logical covers to prejudice the integrity of any clump of beliefs. In effect, we rule out all those logical covers which do not respect clumps. For example, if Simon is unable to start his car one morning, he may come to believe, as the result of inference, that his distributor is at fault. In general, we would be willing to allow such reasoning as correct even though `.BS / D 2 and there exist covering families of the appropriate width which would prevent the inference:  BS X Simon’s distributor is at fault. We would say, in this situation, that the set of Simon’s beliefs about cars and about his car in particular (or at least a respectable subset of these) is immune. The question of which subsets of Bi are to be counted as immune is not to be conclusively settled on logical grounds. This determination must depend upon many factors, not least the identity and circumstances of i.

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165

For ease of exposition and to avoid unnecessary complications, we stipulate that the immune subsets of any premise set have level 1. Our formalization of the intuitive idea of immunity involves a function A, defined on sets of sentences and having as value for a given set €, a set of subsets of € i.e. For all sets € W A.€/  P .€/, where P .€/ is the set of all subsets of € with A.€/ being interpreted informally as the set of all immune subsets of €. In distinction to the definition of †-forcing, we need make no change in the definition of consistency, but rather to the definition of logical cover. To this end we introduce the notion of a logical cover relative to both the underlying logic X and the function A, indicated by COVA X .F; €/: COV A X .F; €/

” ŒCOVX .F; €/ & .8a 2 A.€/9xi 2 F/a  xi 

Thus we allow only those logical covers which do not break up the immune subsets of €. It seems most natural in this setting to consider some single conclusion presentation of the inference relation since we do not usually think of conclusion sets as forming clumps.  AIn either case, however, the definition of the relation we shall call “ X ” will be obvious. There are, of course, many conditions which might be placed upon A-functions, each giving rise to a distinctive sort of A-forcing. Some are of great interest in the studies of modality mentioned earlier in this volume, while others recommend themselves mainly to the inferentially minded. To recover ordinary forcing, we simply set A.€/ empty, i.e.     € ˛.€ / , € A ˛.€ A / & A.€/ D ¿. The other end of this spectrum is not obtained by setting A.€/ D € since this might violate our restriction on the level of immune subsets. The right condition is instead: COV X .A.€/; €/

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(which obviously reduces to A.€/ D € when `.€/ D 1.) This turns out to be one of the modally interesting relations particularly useful in discussions of deontic logic. Under A-forcing we cannot break up immune sets, which will have a nice consequence that we can state if the underlying logic X is productival. [A]  A  A € X ˛1 & : : : & € X ˛k & 9a 2 A.€/ W f˛1 ; : : : ; ˛k g  a  A € X ….˛1 ; : : : ; ˛k / where ….˛1 ; : : : ; ˛k / is the product (in X) of the ˛’s.

9.5 Wrap-up What seems most interesting about forcing and its near relatives is that they actually have room to make some accommodation with the actual world. Think about making changes in classical logic or intuitionism to the same end. It isn’t even clear what changes those might be. Perhaps many-valued logic, as many of its champions have averred, is one of these attempts to meet the real world if not halfway, then at least part of the way. However, forgetting for a moment the deafening howls of protest at such a statement, it is undeniably difficult to see many-valued logic, unless we are talking about a completely Boolean many-valued logic, as any species of classical logic. That is rather the point of classical logic, isn’t it? It is as big as any logic can get over that language, having a Post number of 0.8 So we can’t add anything to classical logic, but neither can we take anything away without the residue being resolutely non-classical. On the other hand. both †-forcing and A-forcing are recognizable as kinds of forcing. Both are defined in terms of level and preserve that property and, at bottom, that is a near definition of what we might call the general notion of forcing. 8 The

Post number of a logic X is the number of non-trivial distinct proper extensions of X.

Ten

Ambiguity Games and Preserving Ambiguity Measures B RYSON B ROWN

Abstract

Brown (1999) applied preservationist ideas to generate consequence relations first exploited by relevance and dialetheic logicians. The central lesson of the paper was that a systematic application of ambiguity can produce consistent images of inconsistent premise sets, allowing us to systematically constrain the consequences that can be inferred from them. Here we present several different ways to apply ambiguity and the preservation of ambiguity measures to obtain paraconsistent logics. The first uses ambiguity to project consistent images of inconsistent premise sets. The second shifts the focus from the syntactic to the semantic, allowing some atoms to receive ambiguous assignments, i.e. assignments that differ from instance to instance. Finally, the third extends the second with a simple game that captures the consequence relation of first degree entailment, dealing symmetrically with triviality on the right as well as the left. The chapter ends with the observation that preservation of a consequence relation under a range of images offers a new way of applying the general theme of preservationism to produce new (and sometimes better behaved) consequence relations.

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168

10.1 Ambiguity Brown (1999) began a line of work that has applied preservationist ideas to generate consequence relations first exploited by relevance and dialetheic logicians. The starting point of this approach was the realization that, by treating certain sets of atomic sentences as ambiguous, we can produce consistent images of inconsistent premise sets. In this chapter we present several different approaches to using ambiguity and the preservation of ambiguity measures to arrive at some familiar consequence relations. The first of these uses ambiguity to project consistent images of inconsistent premise sets. The second shifts the focus from the syntactic to the semantic, allowing some atoms to receive ambiguous assignments, i.e. assignments that differ from instance to instance. The third extends the second with a simple game that captures the consequence relation of first degree entailment. Along the way, dualization of these tricks is applied to cope with triviality on the right as well as the left. We begin with the original, syntactic story of premise sets and their consistent images. A set of formulae, € 0 , is a consistent image of € based on A (which we write ConIm.€ 0 ; €; A/) iff A is a set of atoms, € 0 is consistent, and € 0 results from the substitution, for each occurrence of each member ˛ of A in €, of one of a pair of new atoms, ˛f and ˛ t . A first, crude measure of how far € departs from consistency is the cardinality of the smallest set of atoms A such that for some € 0 , ConIm.€ 0 ; €; A/. But this measure seems to assume that treating an atom as ambiguous carries a fixed, equal cost. We can obtain a more reasonable measure by assuming only that treating any atom as ambiguous carries a non-zero cost. This makes the set of least sets each of which is sufficient for projecting a consistent image of a premise set, € a good measure of €’s inconsistency. So we define the ambiguity set of €: Amb.€/ D fAj9€ 0 W ConIm.€ 0 ; €; A/ ^ 8A0 W A0  A; :9€ 0 W ConIm.€ 0 ; €; A0 /g This is the set of smallest sets, A1 : : : An where for each Ai there is

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169

some €’ such that C onI m€ 0 ; €; Ai . Extensions that require adding more elements to some of the minimal projection bases are unacceptable. However, some acceptable extensions of € will rule out some of these ‘minimal’ projection bases by, in effect, insisting on one or (in general) a proper subset of the bases. For example we might extend f˛, ˛ ! ˇ; :ˇg, whose Amb set is ff˛g,fˇgg by adding ˇ to it; the ambiguity set of f˛, ˛ ! ˇ; :ˇ, ˇg is just ffˇgg, since treating ‘ˇ’ as ambiguous is both necessary and sufficient to produce a consistent image of the original set. So we will regard  as an acceptable extension of € so long as some elements of Amb(€/ are also in Amb(€ [ /: Accept(,€/ , €   & Amb.€ [ /  Amb.€/. Many consequence relations, including the classical consequence relation, can be defined as preserving the acceptability of all acceptable extensions; in this case, our new consequence relation is: € `Amb ˛ , 8: Accept(,€/ ! Accept(f˛g, € [/ It follows that the ambiguity set of the closure of € under `Amb , Amb../CAmb .€; `Amb // D Amb.€/. By including the necessary literals in  we can acceptably extend € to eliminate all but an arbitrary member of Amb(€/. But any `Amb consequence of € must be an acceptable extension of the result. It follows that every minimally costly way of resolving €’s inconsistency remains available when we close € under this consequence relation.

10.2

The Logic of Paradox

Graham Priest’s logic of paradox (LP) is intended to be a dialetheic logic; it is often presented as a simple illustration of such a logic. Priest’s original semantics for LP used Kleene’s strong 3-valued matrices, treating the non-classical value, which is a fixed-point for negation, as designated. But in general, the same consequence relation can be given very different interpretations; in particular, the preservation of ambiguity consequence relation above is identical to the consequence relation of LP. This was originally shown by an induction on the formulae of a

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propositional language. The idea was to establish a one-to-one relation between LP valuations and valuations determined by sets of valuations all of which treat certain atoms ambiguously. To distinguish the new values from the old, we replace T and F in the LP valuations with 1 and 0. The pairs of valuations agree on the atoms assigned true and false, mapping T into 1 and F into 0. But the atoms assigned both by the LP valuation are treated ambiguously. Each instance of each such atom is replaced with one or another of a pair of new atoms, one of which receives the value 1, and the other 0. We quantify across all the resulting projected images, and assign a formula the value 1 if the formula receives the value 1 in some such image. Every sentence assigned either T or B by the LP valuation thus receives the value 1, while the rest receive the value 0. The resulting ambiguity-generated 1/0 valuations are the Scott-valuations1 that determine the LP consequence relation. However, we can give a simpler proof by showing how ‘truth-tables’ for the two logics match up. There are two ways in which an atom can acquire the value 1 in the Scott valuations produced above. It can be assigned the value 1 at the outset, or it can be one of the atoms treated ambiguously (obviously, if it’s one of those, some image of € will substitute the replacement atom assigned 1 for it, and so it receives the value 1 in this second way). Let’s distinguish these two values as 1 and 10 . The difference between them, of course, is just that formulae assigned 1 receive that value in every projected image, while formulae assigned 10 receive 1 in some image and 0 in some image. Now consider how these values combine with : and ^. If a sentence is assigned 1, then its negation is assigned 0 in every image, and so gets the value 0 in the Scott valuation. But if a sentence is assigned 10 , then its negation is also assigned 1 in some image, and so gets the value 1 in the Scott valuation, and the value 10 in our new truth tables. Of course 0 works just as the classical 0 in the tables, so we get, as our new truth table for ‘:’:

1 See

Scott (1974)

10.2. THE LOGIC OF PARADOX

171

ˆ 1 10 0

: 0 10 1

This, of course, is isomorphic to the strong 3-valued Kleene negation. Now we turn to conjunction. Consider , a formula of form  ^ . Once again, we consider the three values, 1, 1’ and 0. If both  and get the value 1, so does ;  gets the value 1’ if one gets the value 1’ while the other gets the value 1, or both get the value 1’. Finally, if either or both get the value 0, so does , of course. So our truth table is ^ 1 10 0

1 1 10 0

10 10 10 0

0 0 0 0

Again, the table for ^ is isomorphic to the table for the strong 3valued Kleene conjunction. The other connectives can be defined in the usual way. It follows that our ambiguity-based consequence relation is identical to LP’s. Another way to present this preservationist reading of the LP consequence relation focuses on the semantic side, rather than on images of the premise set. Wildcard valuations allow inconsistent sets of sentences to be ‘satisfied,’ by treating a set of ’wild-card’ atoms in a way that allows ambiguity. Let L be a propositional language with ˛0 , ˛1 , ˛2 ,. . . the atoms of L, and 1 ,. . . n ,. . . the formulae of L. A wildcard valuation begins by selecting a set of atoms, W , to be the wild cards. We assign values to the sentences of L first by assigning of 0 or 1 uniformly (that is, in the usual way) to each atom not in W . Call this assignment AAt W . Next, we assign 0 or 1 to each instance of an atom in W in each formula of L.2 We call the resulting assignment 2 This

allows each instance of a wild-card atom in each sentence of L to receive either value

CHAPTER 10. PRESERVING AMBIGUITY MEASURES

172 WA At W .

From here, we assign 0 or 1 to each complex formula, using the usual truth functional interpretation of the connectives. The result is a wildcard valuation, W VAt W . Let VAt W be the set of all such valuations based on a given AAt W .3 We quantify across VAt W to obtain a more stable valuation based on all the wildcard valuations for each wildcard set W: Let VA t W be the valuation determined by this quantification over the members of VAt W . Then VAt W 2 L ! f1; 0g, where VAt

W .S /

D 1 if 9V 2 VAt

VAt

W .S /

D 0 else.

W

W V .S/ D 1.

Finally, we define our consequence relation in the usual way: € `W ˛ , 8VAt

W Œ.8

2 €; VAt

W . /

D 1/ ) VAt

W .˛/

D

1.

10.3

Restoring Symmetry

LP is inelegant for the same reasons that singleton forcing is inelegant: It copes with inconsistency on the left, but leaves the dual triviality on the right untouched. In LP, classical contradictions on the left don’t trivialize, but classical tautologies on the right do: Any such tautology follows from every premise set. Further, in its multiple-conclusion form LP eliminates the triviality of inconsistent premise sets, but trivializes all conclusion sets whose closure under disjunction includes a tautology. First degree entailment (FDE) is a closely related logic that treats inconsistency on the left and its dual on the right symmetrically, eliminating the trivialization found in classical logic altogether. (?) presents an ambiguity-based account capturing the consequence relation of first degree entailment. freely. 3 So V At

;

is just the singleton set of one classical valuation on L.

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173

This treatment of FDE requires careful development of the symmetries of the consequence relation. The first step towards re-establishing the symmetries of classical logic in our ambiguity semantics for LP is to dualize the property to be preserved. In the first step we used ambiguity to project consistent images of premise set. Now we will use ambiguity to project consistently deniable images of conclusion sets. Let Amb*(/ be the set of minimal sets of sentence letters whose ambiguity is sufficient to project a consistently deniable image of . We require that any sentence from which  follows be an acceptable extension of every acceptable extension of , where acceptability is consistent deniability: € is an Amb*(/-preserving extension of  , Amb*( [ €/  Amb*(/ Which we write, Accept*(€,/. So a set € is acceptable as an extension of a commitment to denying  if and only if extending  with € doesn’t make things worse, i.e. does not require any more ambiguity to produce a consistently-deniable image than merely denying  does. This leads to a right-to-left consequence relation in which individual sentences are placed on the left and sets of sentences on the right:

`Amb  , 8: Accept*(€,/ H) Accept*(€[f g,/ In English, a set  follows from a formula if and only if is an acceptable extension of every acceptable extension of , a set we are committed to denying. These two asymmetrical consequence relations can be combined to form a symmetrical one: Simply treat sets on the left as closed under conjunction and sets on the right as closed under disjunction, and demand that both these consequence relations apply: € `Sym  , 9ı 2 Cl(, _/: € `Amb ı & 9 2 Cl(€, ^/:

`Amb . Or, by linking the set-set relation to an underlying formula-formula

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relation, we can put it this way instead: € `Sym  , 9ı 2 C., _/, 9 2 Cl(€, ^/: `Sym ı. Where `Sym ı , f g `Amb ı & `Amb fıg. The upshot of this maneuver is a logic sometimes called K*. The consequence relations of FDE and K* agree except when classically trivial sets appear on both the left and the right. In those cases the triviality of the set on the other side ensures that the property we’re preserving on each side is trivially preserved. So K* trivializes when classically trivial sets appear on both the left and the right. FDE demands a little more subtlety. The trick is to produce consistent images of premise sets and nontrivial images of conclusion sets simultaneously, while requiring that the sets of sentence letters used to project these images be disjoint4 . Then € `FDE  iff every such consistent image of € can be consistently extended by some member of each each non-trivial image of  based on a disjoint set of sentence letters, or (now equivalently): € `FDE  iff every such non-trivial image of the conclusion set can be extended by some element of each non-contradictory image of the premise set while preserving its consistent deniability.

10.4

Another Approach to FDE

First degree entailment is, of course, the base of relevance logics, which add to it different accounts of the relevant conditional. The standard semantics for FDE due to Dunn adds two new truth values called ‘both’ and ‘neither,’ to the two familiar ones. In this section we present a new preservationist reading of the job these two values do. On this reading, they express two different treatments of ambiguity in a way that can be captured in a simple game. 4 In effect, ambiguity allows us to capture the results of using ‘both’ and ‘neither’ as (respectively) designated and non-designated fixed points for negation, while insisting that the two sets of ambiguously-treated letters be disjoint ensures that we never treat the same sentence letter in both these ways.

10.4. ANOTHER APPROACH TO FDE

175

The value ‘both’ is a designated fixed-point for negation; similarly, the value neither is a non-designated fixed-point for negation. The consequence relation for FDE preserves Both-or-True from left to right, or (what is the same, in a multiple-conclusion setting) Neither-or-False from right to left. Consider the following recursive definition of a Dunn-valuation, defined on a classical sentential language. We use lower-case Greek letters as variables ranging over the formulae of the language. The definition applies a simple way of identifying Dunn’s four values with the subsets of the set of traditional truth values, fT,Fg: Both= fT,Fg, True=fTg, False=fFg and Neither=;. An assignment to the atoms, Vat 2 At ! fBoth,True,False,Neitherg is extended to the rest of the language according to the following rules: 1. F 2 V(:A) iff T 2 V(A) 2. T 2 V(:A) iff F 2 V(A) 3. F 2 V(A ^ B) iff F 2 V(A) or F 2 V(B) 4. T 2 V(A ^ B) iff T 2 V(A) and T 2 V(B) 5. F 2 V(A _ B) iff F 2 V(A) and F 2 V(B) 6. T 2 V(A _ B) iff T 2 V(A) or T 2 V(B) Note that T 2 V(A) entails that A receives a designated value (either Both or True), and the consequence relation preserves this from left to right, while preserving (in a symmetrical, set-set consequence presentation) Neither or False from right to left. We will prove that our new, symmetrical ambiguity logic is FDE by comparing truth tables, as we did for LP. The truth tables we use emerge from the game mentioned at the beginning of this chapter. The result shows that the work of a Dunn valuation can be done by a game that has nothing to do with peculiar truth-values, because we can arrange the results of the game in tables isomorphic to Dunn’s 4-valued tables for FDE. The game has two players named ‘Verum’ and ‘Falsum.’ It begins with a formula and a partial classical valuation, which matches that to

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the atoms assigned T or F in the corresponding Dunn valuation. Atoms that have not received a classical value are divided between the players. Verum receives all the atoms assigned the value Both in the Dunn valuation, while Falsum receives all those assigned the value Neither. The game is over a selected formula of the language. For Verum, the object of the game is to assign classical values to the instances of her atoms in such a way that the whole formula receives the value T. For Falsum, the object is to assign classical values to all the instances of his atoms in such a way that the whole formula receives the value F. Each player is free to assign either classical value to each distinct instance of any atom they’ve been assigned. This game is either a won game for Verum or a won game for Falsum. If it’s a won game for Verum, then the Dunn valuation assigns either true or both to the formula. If it’s a won game for Falsum, the Dunn valuation assigns either false or neither to the formula. Moreover, the Dunn valuation assigns the value true to the formula if and only if the game is won for Verum even if she and Falsum exchange their assigned letters instances; similarly, it assigns the value false to the formula if and only if the game is won for Falsum even if he and Verum exchange their assigned letters. The preservationist consequence relation based on this game preserves won games for Verum from left to right, or, equivalently, won games for Falsum from right to left. That is, if every game on the left is won for Verum, some game on the right is also won for Verum. We represent each game as a triple, < ; Vc ; D >, where  is the formula, Vc is the partial classical assignment and D 2 (At – Atc / ! fVerum,Falsumg assigns the atoms not assigned a value by Vc to either Verum or Falsum. A game can be won in two ways: In the first case, Vc together with the division of the remaining atoms fixes which of Verum and Falsum wins regardless of which set of atoms is given to Verum and which is given to Falsum; in the second, the win depends also on which of the two sets of atoms is assigned to which player. Proof: By induction on the number of connectives in . 1. Base: For the atoms it’s trivial. The atomic game is won for Verum (Falsum) in the first way if and only if the partial classical valua-

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177

tion assigns it the value T (F). It’s won for Verum (Falsum) in the second way if the atom is assigned to Verum (Falsum). 2. Induction hypothesis: This hold for formulae with up to n connectives. 3. Induction Step: Let  be a formulae with n+1 connectives. We will deal with : and ^ here; the rest of the connectives can be defined in terms of these in the usual way. 4.  has the form : : We have four cases to consider. First, suppose that the game is won for Verum in the first way. Then the game resulting from exchange of letters between Verum and Falsum is also won for Verum. This implies that Falsum can force the value T on by some assignment of T and F to his letters. But this is exactly what Falsum must do to ensure  receives the value F. So  is won in the first way for Falsum. Second, suppose that the game is won for Verum in the second way. Then the game resulting from exchange of letters between Verum and Falsum is won for Falsum. That is, the letters assigned to Verum would allow Verum to force the value F on . But this is exactly what Verum must do to assign the value T to . So  is also a won game for Verum. The third and fourth cases parallel these two, with Verum and Falsum trading places, so if is won for Falsum in the first way,  is won for Verum in the first way, and if is won for Falsum in the second way,  is won for Falsum in the second way as well. 5.  has the form ^ : Here we have eight cases to consider, since both and  can be won in either way for either player. However, the cases go through very straightforwardly. First, suppose both and  are won for Verum in the first way. Then Verum can force the value T on  whether she plays her own atoms or Falsum’s atoms—that is, the A game is won for Verum in the first way as well. But if either or both of and  is won for Verum in the second way, then Falsum wins the  game on an exchange of letters, and  is won for Verum in the second way. If either or both of

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and  is won for Falsum in the first way, then  is won for Falsum in the first way, while if both and  are won for Falsum in the second way,  is won for Falsum in the second way.  We can summarize the results of this theorem in a table:  won 1 for Verum won 2 for Verum won 2 for Falsum won 1 for Falsum

: won 1 for Falsum won 2 for Verum won 2 for Falsum won 1 for Verum

^ won 1 for Verum won 2 for Verum won 2 for Falsum won 1 for Falsum

won 1 for Verum won 1 for Verum won 2 for Verum won 2 for Falsum won 1 for Falsum

won 2 for Verum won 2 for Verum won 2 for Verum won 2 for Falsum won 1 for Falsum

^ won 1 for Verum won 2 for Verum won 2 for Falsum won 1 for Falsum

won 2 for Falsum won 2 for Falsum won 2 for Falsum won 2 for Falsum won 1 for Falsum

won 1 for Falsum won 1 for Falsum won 1 for Falsum won 1 for Falsum won 1 for Falsum

These tables are isomorphic to Dunn’s four-valued tables for FDE: W on1 , for Verum and Falsum respectively, corresponds to true and to false, while W on2 similarly corresponds to both and to none. So our logical game between Verum and Falsum does the work of a Dunn valuation. This trick provides yet another way of applying ambiguity to replace strange truth values—the rules of the game allow Verum and Falsum to treat the atoms assigned to them ambiguously, as they attempt to produce an assignment that makes the target formula true or false, but

10.5. AMBIGUITY AND QUANTIFICATION

179

each player uses the leeway that ambiguity grants her in a particular way.

10.5 Ambiguity and quantification Recall the wild card atoms considered above as a treatment of LP. Our little game shows that we can also replace the values both and none with wild cards. For each Dunn valuation, we consider the corresponding wild card valuations. These are wild-card valuations such that: 1. Atoms to which the Dunn valuation assigned T and F are assigned the values 1 and 0 respectively. 2. The rest of the atoms are treated as wild cards, divided between the atoms assigned both by the Dunn valuation and those assigned none. 3. If some assignment of T, F to instances of the atoms assigned Both by the Dunn valuation forces the value T on the formula, whatever values are assigned to the instances of atoms assigned none by the Dunn valuation, we also assign 1 to the formula. 4. Otherwise we assign 0 to the formula. The resulting 1,0 valuations each correspond to a Dunn valuation, assigning 1 to every formula assigned a designated value by the Dunn valuation and 0 to every formula assigned an undesignated valuation. So the consequence relation determined by our game also results from considering all the possible assignments to instances of the wild card atoms, and looking for whether a ‘win’ for Verum, an ambiguous assignment to the atoms assigned Both by the Dunn valuation that forces truth on the formula, exists among them.

10.6 Echoes of supervaluation Our methods here bear a resemblance to Bas van Fraassen’s supervaluations. A supervaluation also begins as a partial classical valuation, extends that valuation by assigning values to the ‘gaps’ in every way, and

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then quantifies across all the resulting extended valuations, conservatively assigning T (F) to a formula  only if all the extended valuations assign T (or F) to . Our ambiguity logics retain both the starting point, a partial classical valuation, and the use of quantification across a range of extended valuations to compensate for the arbitrariness of the extensions of that partial valuation. These devices allow us to achieve our results without adding strange truth values to the mix, just as they do for van Fraassen. By adding the further element of ambiguity, we have been able to capture paraconsistent consequence relations that resist the trivialization of inconsistent sets on the left and of sets which are not consistently deniable on the right.

10.7

Final Remarks on Preservation

There is another way to express what is preserved by these logics, which opens up a broader understanding of preservation. This approach focuses on preserving the consequence relation itself. We can say that the classical consequence relation is preserved here, under a range of minimally ambiguous, consistent (or consistently deniable) images of our premises and conclusions: € `FDE  iff every image of the premise and conclusion sets, I.€/, I  ./ obtained by treating disjoint sets of sentence letters as ambiguous is such that I.€/ ` I  ./. This suggests a new preservationist strategy for producing new consequence relations from old. We can say that the new consequence relation holds when and only when the old relation holds in all of a range of cases anchored to the original premise and conclusion sets. This strategy eliminates or reduces trivialization by ensuring that the range of cases considered includes some non-trivial ones, even when the case forming our ‘anchor’ is trivial. This idea can also be applied to the weakly aggregative forcing relation; the ambiguity-based treatment of forcing allows ambiguity at the level of formulas, but not within formulas, and uses a maximum number of distinct ‘colors’ corresponding to the levels of the premise and

10.7. FINAL REMARKS ON PRESERVATION

181

conclusion sets. The familiar forcing consequence relation can then be presented not as preserving levels of incoherence but as preserving the classical consequence relation itself, under quantification across a range of recolorings of the elements of premise and conclusion sets.

Nomenclature 1; 0; 10 The three values of LP for Scott-valuations, page 170 A AAt

A set of atoms, page 168 W

The pre-wildcard assignment, page 171

H

The domain of a meaning function, page 105

I.€/

A consistent image of €, page 180

L

The set of sets which preserve level by union, page 131

M

The class of covers, page 109

MX

The class of models of an algebraic logic, page 104

Mod.†/ The class of models of the set †, page 107 RK3

The canonical frame relation for K3, page 8

Rxyz The objects x; y and z stand in the relation R, page 4 T H.x/ the set of all minimal transversals for H.x/, page 50 T h.K/ The theory of the class of models K, page 106 VAt

W

The set of wildcard valuations based on AAt

W,

page 172

W on1 The game-theoretic truth value for true and false, page 178 W on2 The game-theoretic truth value that corresponds to the truth values ‘both’ and ‘none’, page 178 183

NOMENCLATURE

184

X; Y

Variables for logics, page 84

¿

The empty set, page 85

Amb.€/ The Ambiguity set of €, page 168 C

Variable for covers, page 108

Cm €

The minimal cover of €, page 110

ConIm.€ 0 ; €; A/ The set € 0 is the consistent image of € with respect to the set of atoms A, page 168 L

The language of a logic, page 84

F

A variable for a logical cover of the set kind, page 90

€; †

sets of formulas, page 6

€

 Set difference, page 108

€C

An MLPE , page 131

M; R Models of a logic, page 107 MK3 The canonical model of K3, page 8 -MLPE An -consistent like MLPE , page 134 `

The syntactic or proves relation, page 5

˛; ˇ; ; ' Formula variables, page 4


The (n-1)-ary modal operator, page 5

CON X

Consistency predicate for the logic X, page 85

[ 

Set union, page 85

`X

The level function of the logic X, page 110

`X

The level function relative to a logic X where the covers are sets, page 91

The syntactic level-forcing relation, page 11

NOMENCLATURE

185

!

The finite subset of relation, page 126 W 2 1i<j nC1 ˛i ^ ˛j for sentences ˛1 ; ˛2 ; :::; ˛nC1 , nC1 .˛i /i2ŒnC1 page 56 S 2 1i<j nC1 ti \ tj for sets t1 ; t2 ; :::; tnC1 , page 56 nC1 .ti /i2ŒnC1



The material conditional., page 5

H)

The meta-language conditional, page 11

1

The special value of level functions for absurd formulas, page 91



The ‘less than or equal to’ relation on the integers, or a general relation of partial order, page 129

L

a modal logic, page 51

Nq

The set of axioms for hyperframes, page 50

H

hypergraph function, page 48

U

universe of a hyperframe, page 48

M

a hypermodel, page 48

F

Varible for a hyperframe, page 48

ˆX

The satisfaction relation of an algebraic logic, page 104



The necessity operator from modal logic, page 5

:

The negation operator, page 6

ha; bi The ordered pair of a and b, page 4 f˛1 ; : : : ; ˛n g The set of ˛1 ; : : : ; ˛n , page 20 !

The classical conditional (alternative), page 169



Semantic consequence, page 105

B3

The K3 provability relation, page 7

NOMENCLATURE

186

`W

The ‘wildcard’ consequence relation, page 172

`Amb The ambiguity set consequence relation, page 169 _

The ‘or’ operator, page 122

^

The ‘and’ operator, page 6

WA At W

The wild card assignment, page 172

WV At W

The wildcard valuation, page 172

f W A ! B In general, f is a map from A to B, page 105 mngX The meaning function of a logic X, page 105 w.C/ The width of the cover C, page 108 [Cut-1] A version of [Cut], page 108 [Cut] The structural rule of inference of called [Cut], page 85 [Den*] The relation of Symmetric Negation Denial, page 122 [Den] The property that denial commutes with provability, page 87 [K3]

The aggregation rule for the modal logic K3, page 6

[K]

The rules of modal aggregation, page 5

[M]

The rule of monotonicity (alternative), page 108

[Mon] The structural rule of inference: monotonicity, page 85 [N]

The rule of necessitation, page 6

[RM] The rule of distribution of the  operator over the  connective, page 5 [R]

The structural rule of inference of reflexivity, page 85

MLPE

Maximal level preserving extension, page 131

[M ] The rule of level preserving monotonicity for forcing, page 125

NOMENCLATURE

[Cut ] The special version of [Cut] for forcing, page 125 Accept(,€/  is an acceptable extension of €, page 169 Cn

The set of connectives of a logic, page 105

FDE

First degree entailment, page 172

IL

Intuitionistic Logic, page 122

LP

Priest’s Logic of Paradox, page 102

ND

Negation Denial, page 121

187

References Apostoli, P. and B. Brown (1995a). A solution to the completeness problem for weakly aggregative modal logic. Journal of Symbolic Logic 60, 832–842. Apostoli, P. and B. Brown (1995b). A solution to the completeness problem for weakly aggregative modal logic. Journal of Symbolic Logic 60, 832–842. Avron, A. (1994). What is a logical system? In What is a logical system?, Volume 4 of Stud. Logic Comput., pp. 217–238. New York: Oxford Univ. Press. Blackbutn, P., M. de Rijke, and U. Venema (2001). Modal Logic. Cambridge. Brown, B. and P. K. Schotch (1999). Logic and Aggregation. Journal of Philosophical Logic 28(396), 265–287. Campbell, R. (1980). Can inconsistency be reasonable? Journal of Philosophy 11(2), 245–270.

Canadian

Chellas, B. (1980). Modal logic: an introduction. Cambridge. Davey, B. and D. Duffus (1982). Exponentiation and duality. In I. Rival (Ed.), Ordered sets, pp. 43–95. Reidel. Davey, B. and H. Priestley (1990). Introduction to lattices and order. Cambridge: Cambridge University Press. 189

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Fitting, M. (1986). Notes on the mathematical aspects of Kripke’s theory of truth. Notre Dame J. Formal Logic 27(1), 75–88. Jennings, R. and P. Schotch (1981a). Some remarks on Weakly weak modal logics. Notre Dame Journal of Formal Logic 22, 309–314. Jennings, R. and P. Schotch (1981b). Some remarks on (weakly) weak modal logics. Notre Dame Journal of Formal Logic 22, 309–314. Jennings, R. and P. Schotch (1984). The preservation of coherence. Studia Logica 43(1/2), 89–106. Johnston, D. (1978, February). A generalized relational semantics for modal logic. M.A. thesis, Simon Fraser University, Burnaby, British Columbia, Canada. J´onsson, B. and A. Tarski (1951). Boolean algebras with operators. Part I. Amer. J. Math. 73, 891–939. J´onsson, B. and A. Tarski (1952). Boolean algebras with operators. Part II. Amer. J. Math. 74, 127–162. Kripke, S. A. (1963). Semantical considerations on modal logics. Acta Philosophica Fennica, Modal and Many-Valued Logics 16, 83–94. McKinsey, J. and A. Tarski (1948). Some theorems about the sentential calculi of Lewis and Heyting. Journal of Symbolic Logic 13, 1–15. Nicholson, T., R. Jennings, and D. Sarenac (2000a). Revisiting completeness for the Kn modal logics: a new proof. the Logic Journal of the IGPL 8(1), 101–105. Nicholson, T., R. E. Jennings, and D. Sarenac (2000b). Revisiting completeness for the Kn modal logics: A new proof. Logic Journal of the IGPL 8, 101–105. Payette, G. and B. d’Entremont (2006, December). Level Compactness. Notre Dame Journal of Formal Logic 47(4), 545–555. Payette, G. and P. K. Schotch (2006, Feb). On Preserving.

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Priestley, H. (1970). Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190. Priestley, H. (1984). Ordered sets and duality for distributive lattices. Annals of Discrete Mathematics 23, 39–60. Quine, W. V. O. and J. S. Ullian (1970). The Web of Belief (2nd (1978) ed.). McGraw Hill. Schotch, P. and R. Jennings (1980a). Inference and necessity. J. of Philosophical Logic 9, 327–340. Schotch, P. and R. Jennings (1980b). Inference and necessity. Journal of Philosophical Logic IX, 327–340. Schotch, P. and R. Jennings (1980c). Inference and necessity. Journal of Philosophical Logic 9, 327–340. Schotch, P. and R. Jennings (1980d). Modal logic and the theory of modal aggregation. Philosophia 9, 265–278. Schotch, P. and R. Jennings (1980e). Modal logic and the theory of modal aggregation. Philosophia 9, 265–78. Schotch, P. and R. Jennings (1989). On detonating. In G. Priest, R. Routley, and J. Norman (Eds.), Paraconsistent Logic, pp. 306–327. Philosophia Verlag. Schotch, P. K. (2004). An Introduction to Logic and its Philosophy (0 ed.). N/A. Scott, D. (1974). Completeness and axiomatizability in many-valued logic. In Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971), Providence, R.I., pp. 411–435. Amer. Math. Soc. Segerberg, K. (1971). An Essay in Classical Modal Logic. Uppsala. Stone, M. (1936). The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111.

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Tarski, A. and B. J´onsson (1951). Boolean algebra with operators i. American Journal of Mathematics 73, 891–939. Urquhart, A. (1995). Completeness for weakly aggregative modal logic. manuscript.

Index X-level forcing; axiomatization, 139 urelementen, 84 ‘A and not-A’ capable of being true, 29

compactness, 84 complete modal aggregation, 5 compositional; definition of, 106 compositionality, 103 conclusion sets, 172–174, 181 conjunction, 171; of conclusions, 92 connective, 105; derived, 121 connectives, 171, 172, 177 consistency, 84; definition of, 85; definitionpreservation of, 85, 86; inadequacy of classical account, 15 ; predicate, 85; preservation in the strong sense, 86; preservation of, 85, 88 consistent, 114 consistent deniability, 174 consistently deniable, 173 constant symbol, 109 contradictions; implicit vs explicit, 22 corresponding conditionals, 82 cost; of treating atoms as ambiguous, 168 Costa-Leite, Alexandre, 103 cover, 108; intrepreted as a model, 109

absurd, 88, 91, 110; (ity), 102; formula, 92 ; formulas, 10, 20, 91 acceptable extension, 169, 173 aggregation principle; for ternary modal logic, 6 alternative inferential lifestyle, 25 ambiguity measures, 168 ambiguity set, 168 ambiguous; assigments, 168; sets of atomic formulas, 168 anti-symmetry, 129 Apostoli, Peter, 47 arithmetic, 24 atomic formula, 168, 170 Avron, Avrin, 102 beliefs; active, 22; attempt to deny inconsistency of, 25; drawing inferences from, 22; real vs. apparent, 25 Boolean Algebras, 5 Brown, Bryson, 13, 47, 109

d’Entremont, Blaine, 13, 125, 126 Dalhousie University, 3 deducibility; semantics of implication, 18 deductive closure, 83 denial, 87; contingent, 121; negation-, 121 ; non-trivial, 89, 121; symmetric negation-, 122 dialetheic, 168 dialethism, 29 dialethists; and truth-preservation, 12 dilution, see monotonicity disjoint, 174 doctrine of original sin, 25

calculi, 122 cardinality, 110, 168 category theory, 132 chain; in a partially ordered set, 129 Chellas, B., 6 Chellas, Brian, 48 chromatic compactness, 48 classical consistency; as an ideal, 22 coloring; k-colourable, 48; compactness for, 48

193

INDEX

194 dualization, 168 Dunn, M., 174 Dunn, Michael, 175 endomorphism, 106, 107 entailment, 105 ex falso quodlibet, 19, 102 explicit contradiction, 23; as a single sentence, 23; believing an, 23 Falsum, 175 fixed point; for negation, 175 fixed-level forcing relations, 10 forcing, 103, 107, 115, 180 ; X-level, 98 ; A-forcing, 96 ; closure relative to X, 123; collaspes into the underlying logic, 123; definition of, 96; fixed-level version of, 12; is not monotonic, 124; level-forcing, 100; preserves level, 12; relation (not Cohen’s version), 11 formalism, 17 formula, 107; absurd, 102, 121; algebra of. . . (s), 107; complex, 172; contingent, 121; denial of a, 121; schema, 122 frame relation, 7; binary, 4; diagonal fragment, 5; n-ary, 4; relaxing condition on, 4; ternary, 4 function, 106 gaggle-truth; vs single formula truth, 83 Galois Connected, 7 game, 174, 175 general moral principle, 26 Goethe, 5 Halifax, 7 homomorphic, see homomorphism homomorphism, 106; meaning functions are, 106 Hume, 21, 22 Hume’s Labyrinth, 21 hyperconsistent, 92 hyperframe; n-bounded, 53; canonical frame, 51; simple, 50 hyperframes, 48, 53; is axiomatized by, 50 hypergraph; function, 48 hypergraphs, 48; transverse, 48

hypermodel, 48 implicit contradiction ; as mutually inconsistent sentences, 23 incoherence, see inconsistency inconsistency, 89, 102; need to distinguish two kinds, 20 inconsistent, 94; definition of, 85; premise sets, see premise sets; relative to forcing, 98 inconsistent belief sets; two approaches, 22 individual quantifiers; a needed revision of classical logic, 16 inference, 97; from inconsistent sets, 94; structural rules of, 107 inference relation, 84; paraconsistent, 9 intuitionists, 16 J´onsson, B., 33 J´onsson, B, 5 J´onsson, B., 5, 31 Jennings and Schotch, 5 Jennings, Ray, 3–109; Jennings’ crucial insight, 7 K3 modal logic, 7, 9 ; canonical model, 8 ; determined by the class of all ternary frames, 9; K3-canonical relation, 8; maximal K3-consistent set, 8 Kleene, S., 169, 171 Kripke, S., 5 left-handed sets, 20 Leung, Kam Sing, 13 level, 89, 108; -forcing, 111; as a measure of inconsistency, 110 ; definition of, 91; is compact, 130; is downward monotonic, 123; monotonicity of, 127; of a set, 111 ; of a set of formulas, 11; of a set of theorems, 111 ; of consistency, 92; of inconsistency, 91; preservation of, 92; transfinite, 124 level preservation; fixes truth-preservation, 12 Lewis, C.I., 13, 16–18, 20 literal(s), see asoatomic formula169 Logic of Paradox, 102, 169

195 logic ; 3-valued, 102 ; algebraic, 103, 105 ; ambiguity, 180; at evens, 87 ; Classical, 101 any logic is a sublogic of, 16 as a recent invention, 17 as an entrenched theory, 17 ; compact, 124; compositional, 106 ; definition of, 84 ; extension view of, 84; first-order, 105, 130; inconsistency intolerant, 137 ; intuitionisitic, 134; Intuitionistic as the most successful replacement theory, 17; intuitionistic, 101, 122; modal, see modal logic ; non-classical, 24, 25; non-monotonic, 25 ; non-normal, 52 ; paraconsistent, see paraconsistent logic, 123 ; propositional, 16 ; structural, 108 ; substitutional, 113 ; trivial/non-trivial, 85 logical cover, 90; cells of a, 96; conditions on equality, 109; narrowest, 91 ; partition, 90, 114; relative to a logic X, 90 ; width of, 91, 108 logical pluralism, 101 logical truths; their privileged status?, 19 lottery paradox, 95 many-valued logics, 16 map, 106 material implication, 16 maximal level preserving extension, 128 meaning function, 105; definition of, 105 metalanguage; concept not accepted by Lewis, 19; metalinguistic level, 19 metaphysics, 21 Minas, J.S., 14 minimal models, 48 minimal transversal, 50 modal aggregation, 6 modal logic, 51 ; axiomatized by the rule [N], 6; n-ary, 7; as disreputable, 18 ; frame conditions, 4 ; normal, 3, 4 ; rule of monotonicity, 5; rule of normality or necessitation, 5 ; the logic K, 7 ; weakly aggregative, 55

modal semantics; and paraconsistency, 7 Modus Morons; rule of, 24 monotone operator, 129 monotonicity, 108; level preserving, 125; levelpreserving, 98, 100; unrestricted, 97 moral argument, 26 moral dilemma, 26, 28 more humanistic logic, 21 Morgan, C., 4 morphism, see also map and function necessity ; characterized in terms of inference, 7 neighborhood semantics, 48 Nicholson, Dorian, 13, 32 Nicholson, Dorion, 48 non-adjunctive; slighting term for ‘preservationist’, 10 non-negative; slighting term for ‘dialethic’, 10 not knowing what X is; Quinean sense, 19 obligations, 89 ; closed under logical consequence?, 26; conflict of, 26, 28 ordinal, 108, 111 ordinary doxastic agents, 21 ordinary moral experience, 26 paraconsistent, 102, 104 paraconsistent logic, 3, 9–10, 96; dialethic approach to, 10; first heard of, 6 paradoxes of material implication, 18 Partee, B., 6 partially ordered set, 129 partially paraconsistent inference relation, 9 Payette, Gillman, 13, 125, 126 pigeonhole reasoning, 6 Pittsburgh, 3 possible worlds, 4 Post, Emil; and the completeness of classical logic, 16 premise set; consistent image, 168 premise sets, 168 ; some cannot be partitioned into consistent subsets, 10; syntatic story of, 168; triviality of, 172

INDEX

196 preservationism, 7, 83; origins, 3 preservationist, 171, 174, 176, 180 Priest, Graham, 10, 102 Priest,Graham, 169 Priestley,H., 31 prime number, 84 probability calculus, 25 productival, 100, 132 projection bases, 169 provability relation, see inference relation Quine, W.V.O., 18, 19, 82, 95 rationality; our ordinary understanding, 22 reductio; sometimes wrongly invoked, 20 reasoning from inconsistent data, 6 reflexivity, 107 n C 1-ary relational frames, 53 relevance of logic to ethics, 27 right-handed sets, 20 Russell, B., 18 satisfaction relation, 104 Schotch, Peter, 3–122 Scott, D., 6, 170 Scott-valuations, 170 Segerberg, Krister, 48 self-contradictory premise, see absurd formulas semantics; consequence with respect to, 105 Simon Fraser University, 3 singular set of premises, 98 Society for Exact Philosophy, 3, 6 strict implication, 16, 18, 19; is just provability, 19; Quine’s criticism improved, 19 structural; rules of inference, 85 substitution; level preserving, 116 substitution property, 107; weak and strong versions of, 107 supervaluations, 179 symbolic value 1; as value of level function, 11 Tarski, A., 5, 31, 33 tautology, 172 ; as dual-self-contradiction, 20 The Lewis position ; applied to belief, 24 ; Quinean criticisms, 18; recast

as a complaint about provability, 19 theorem, 107, 121 theory, 106; as set of formulas, 83 true contradictions; counterintuitive, 29; hallmark of dialethism, 10 true sets; must be stipulated, 84 truth ; coherence account, 83 ; defined for single sentences, 84; on a polyadic frame, 53; predicate of sentences, 83 truth value assignment; partial, 179 truth values; ambiguity replaces strange, 178 truth-condition for necessity; in ternary frames, 4 truth-maintenance systems, 95 truth-preservation, 82; and para-consistency, 11; defined, 81; transmuting rather than preserving truth, 83; two kinds of truth used, 82 truth-table, 170 truth-value; classical, 176; designated, 169, 179; Dunn’s 4-truth values, 174 ; non-classical, 169 two strategies for fixing a logic, 15 uniform substitution, 107 Urquhart, Alasdair, 13 Urquhart, Alisdair, 47 valid argument, 25 van Frassen, B., 179 variable; free, 109 Venn diagrams, 24 Verum, 175 wild card atoms, 179 wildcard valuations, 171 Wolf, R., 6

Contributors Bryson Brown is Professor of Philosophy at the University of Lethbridge. Raymond Jennings is Professor of Philosophy at Simon Fraser University. Kam Sing Leung is a recent graduate of the doctoral program of the Philosophy Department of Simon Fraser University. Dorian Nicholson is a recent graduate of the doctoral program of the Philosophy Department of Simon Fraser University. Gillman Payette is in the doctoral program of the University of Calgary and the recipient of the Governor General’s gold medal for his graduate work at Dalhousie University. Peter Schotch is Professor of Philosophy at Dalhousie University. Alasdair Urquhart a Professor of Philosophy and of Computing Science at the University of Toronto.